| L(s) = 1 | + 6·5-s − 10·13-s − 6·17-s − 59·25-s + 138·29-s − 20·37-s + 108·41-s + 155·49-s + 252·53-s − 14·61-s − 60·65-s + 74·73-s − 36·85-s + 168·89-s + 20·97-s + 630·101-s − 148·109-s + 138·113-s + 572·121-s − 198·125-s + 127-s + 131-s + 137-s + 139-s + 828·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 6/5·5-s − 0.769·13-s − 0.352·17-s − 2.35·25-s + 4.75·29-s − 0.540·37-s + 2.63·41-s + 3.16·49-s + 4.75·53-s − 0.229·61-s − 0.923·65-s + 1.01·73-s − 0.423·85-s + 1.88·89-s + 0.206·97-s + 6.23·101-s − 1.35·109-s + 1.22·113-s + 4.72·121-s − 1.58·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 5.71·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(29.48712044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.48712044\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 3 T + 43 T^{2} - 234 T^{3} + 936 T^{4} - 234 p^{2} T^{5} + 43 p^{4} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 - 155 T^{2} + 13885 T^{4} - 853634 T^{6} + 44488354 T^{8} - 853634 p^{4} T^{10} + 13885 p^{8} T^{12} - 155 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 - 52 p T^{2} + 14534 p T^{4} - 29130320 T^{6} + 3978186283 T^{8} - 29130320 p^{4} T^{10} + 14534 p^{9} T^{12} - 52 p^{13} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 5 T + 385 T^{2} + 14 p T^{3} + 69814 T^{4} + 14 p^{3} T^{5} + 385 p^{4} T^{6} + 5 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 3 T + 334 T^{2} + 693 T^{3} + 110178 T^{4} + 693 p^{2} T^{5} + 334 p^{4} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1157 T^{2} + 639226 T^{4} - 232952123 T^{6} + 79410905002 T^{8} - 232952123 p^{4} T^{10} + 639226 p^{8} T^{12} - 1157 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 3179 T^{2} + 4766701 T^{4} - 4443245426 T^{6} + 2817726255202 T^{8} - 4443245426 p^{4} T^{10} + 4766701 p^{8} T^{12} - 3179 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 69 T + 2785 T^{2} - 54630 T^{3} + 1258182 T^{4} - 54630 p^{2} T^{5} + 2785 p^{4} T^{6} - 69 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 5015 T^{2} + 11803225 T^{4} - 17839334522 T^{6} + 19682029431166 T^{8} - 17839334522 p^{4} T^{10} + 11803225 p^{8} T^{12} - 5015 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 10 T + 1720 T^{2} - 76250 T^{3} + 347470 T^{4} - 76250 p^{2} T^{5} + 1720 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 54 T + 4024 T^{2} - 81324 T^{3} + 5546553 T^{4} - 81324 p^{2} T^{5} + 4024 p^{4} T^{6} - 54 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 8420 T^{2} + 34845370 T^{4} - 96102305408 T^{6} + 200231923723651 T^{8} - 96102305408 p^{4} T^{10} + 34845370 p^{8} T^{12} - 8420 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 5963 T^{2} + 23241085 T^{4} - 56915331554 T^{6} + 134207961483970 T^{8} - 56915331554 p^{4} T^{10} + 23241085 p^{8} T^{12} - 5963 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 126 T + 12208 T^{2} - 787698 T^{3} + 46295070 T^{4} - 787698 p^{2} T^{5} + 12208 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 11900 T^{2} + 83484274 T^{4} - 424711340048 T^{6} + 1692217412552155 T^{8} - 424711340048 p^{4} T^{10} + 83484274 p^{8} T^{12} - 11900 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 7 T + 8527 T^{2} - 144110 T^{3} + 35209516 T^{4} - 144110 p^{2} T^{5} + 8527 p^{4} T^{6} + 7 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 - 23588 T^{2} + 286973770 T^{4} - 2219726923904 T^{6} + 11886387397766035 T^{8} - 2219726923904 p^{4} T^{10} + 286973770 p^{8} T^{12} - 23588 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( 1 - 14120 T^{2} + 150082588 T^{4} - 1080637518872 T^{6} + 6319440918414790 T^{8} - 1080637518872 p^{4} T^{10} + 150082588 p^{8} T^{12} - 14120 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 37 T + 20314 T^{2} - 565891 T^{3} + 160126666 T^{4} - 565891 p^{2} T^{5} + 20314 p^{4} T^{6} - 37 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 26759 T^{2} + 385201033 T^{4} - 3770516962586 T^{6} + 27272334530940718 T^{8} - 3770516962586 p^{4} T^{10} + 385201033 p^{8} T^{12} - 26759 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 14351 T^{2} + 189753289 T^{4} - 1826209201322 T^{6} + 13547553864599998 T^{8} - 1826209201322 p^{4} T^{10} + 189753289 p^{8} T^{12} - 14351 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 84 T + 30700 T^{2} - 1886940 T^{3} + 359702982 T^{4} - 1886940 p^{2} T^{5} + 30700 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 10 T + 22648 T^{2} + 300908 T^{3} + 243615193 T^{4} + 300908 p^{2} T^{5} + 22648 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.87928624268030439438380709232, −3.78627217965478827091309600530, −3.70084943311142375437557490887, −3.29548999482464805774278297974, −3.29522077104879942926147822549, −3.26085474118393845473935139383, −3.20830759940494090071619534637, −2.96820355947396250354415397185, −2.82218807818679329187080598434, −2.53958079969694387095550842257, −2.43505964821665577098144568023, −2.24990969289169134525572086587, −2.24070398730213947168921898642, −2.22969047219060243897357917481, −2.04086258748004447228989082756, −1.93165685315087173109146003648, −1.88592827563254384043519654300, −1.42068971871802067016336347031, −1.19020544969034686347134959408, −1.03985134579038815669085441846, −0.794944159684046629880029778781, −0.77756631693632127627152030305, −0.61591712694013038365836341885, −0.58929613348753539769720043803, −0.25045219026863023961232320298,
0.25045219026863023961232320298, 0.58929613348753539769720043803, 0.61591712694013038365836341885, 0.77756631693632127627152030305, 0.794944159684046629880029778781, 1.03985134579038815669085441846, 1.19020544969034686347134959408, 1.42068971871802067016336347031, 1.88592827563254384043519654300, 1.93165685315087173109146003648, 2.04086258748004447228989082756, 2.22969047219060243897357917481, 2.24070398730213947168921898642, 2.24990969289169134525572086587, 2.43505964821665577098144568023, 2.53958079969694387095550842257, 2.82218807818679329187080598434, 2.96820355947396250354415397185, 3.20830759940494090071619534637, 3.26085474118393845473935139383, 3.29522077104879942926147822549, 3.29548999482464805774278297974, 3.70084943311142375437557490887, 3.78627217965478827091309600530, 3.87928624268030439438380709232
Plot not available for L-functions of degree greater than 10.
