| L(s) = 1 | − 8·4-s + 12·9-s − 48·11-s + 40·16-s − 240·29-s − 96·36-s + 384·44-s + 20·49-s − 160·64-s − 240·71-s − 256·79-s + 90·81-s − 576·99-s + 16·109-s + 1.92e3·116-s + 400·121-s + 127-s + 131-s + 137-s + 139-s + 480·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 488·169-s + ⋯ |
| L(s) = 1 | − 2·4-s + 4/3·9-s − 4.36·11-s + 5/2·16-s − 8.27·29-s − 8/3·36-s + 8.72·44-s + 0.408·49-s − 5/2·64-s − 3.38·71-s − 3.24·79-s + 10/9·81-s − 5.81·99-s + 0.146·109-s + 16.5·116-s + 3.30·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 10/3·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.88·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4869430705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4869430705\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} )^{4} \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 20 T^{2} + 6 p^{2} T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \) |
| good | 11 | \( ( 1 + 12 T + 260 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 244 T^{2} + 53574 T^{4} + 244 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 320 T^{2} + 546 p^{2} T^{4} - 320 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 652 T^{2} + 348486 T^{4} - 652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1720 T^{2} + 1275954 T^{4} - 1720 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 30 T + p^{2} T^{2} )^{8} \) |
| 31 | \( ( 1 - 1252 T^{2} + 745926 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 508 T^{2} - 334362 T^{4} + 508 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 4960 T^{2} + 11110434 T^{4} - 4960 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 1748 T^{2} + 228678 T^{4} - 1748 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 3940 T^{2} + 5766 p^{2} T^{4} + 3940 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4828 T^{2} + 18249126 T^{4} - 4828 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 4420 T^{2} + 6539622 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 14596 T^{2} + 80934054 T^{4} - 14596 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 13508 T^{2} + 83688486 T^{4} - 13508 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 60 T + 4484 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 13108 T^{2} + 98972646 T^{4} + 13108 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 64 T + 3138 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 16612 T^{2} + 134415078 T^{4} + 16612 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 8896 T^{2} + 52680354 T^{4} - 8896 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 24820 T^{2} + 317127462 T^{4} + 24820 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−3.94191228401396976259769361943, −3.84384514149266585850965230631, −3.79823434654078036759138241566, −3.77377914386260241166981363021, −3.74402905928937343598455154314, −3.49545673157748434328507090569, −3.14329567416303232623150966748, −3.00417455185109934554264778021, −2.95154787015763153053191718400, −2.83306561183549535993028376963, −2.74845096701236678423249317445, −2.41792520358083221819932736173, −2.38255923385605402318527166481, −2.17929225163690731501125354646, −2.15995765214832425440344256038, −1.72333611964971662373658727119, −1.62143940326523710664262219940, −1.60417878957628496399688387690, −1.38421152977048622487233128098, −1.33939133183138802123136169255, −0.980271013007225842991207609547, −0.35216348598112503728484979510, −0.30817409510763050177991686380, −0.27618175968791770979463784608, −0.26650109519068279188416239077,
0.26650109519068279188416239077, 0.27618175968791770979463784608, 0.30817409510763050177991686380, 0.35216348598112503728484979510, 0.980271013007225842991207609547, 1.33939133183138802123136169255, 1.38421152977048622487233128098, 1.60417878957628496399688387690, 1.62143940326523710664262219940, 1.72333611964971662373658727119, 2.15995765214832425440344256038, 2.17929225163690731501125354646, 2.38255923385605402318527166481, 2.41792520358083221819932736173, 2.74845096701236678423249317445, 2.83306561183549535993028376963, 2.95154787015763153053191718400, 3.00417455185109934554264778021, 3.14329567416303232623150966748, 3.49545673157748434328507090569, 3.74402905928937343598455154314, 3.77377914386260241166981363021, 3.79823434654078036759138241566, 3.84384514149266585850965230631, 3.94191228401396976259769361943
Plot not available for L-functions of degree greater than 10.
