| L(s) = 1 | + 24·4-s + 10·9-s − 44·11-s + 316·16-s − 144·19-s − 288·29-s − 68·31-s + 240·36-s + 1.07e3·41-s − 1.05e3·44-s + 1.07e3·49-s − 1.26e3·59-s + 1.68e3·61-s + 2.88e3·64-s − 1.35e3·71-s − 3.45e3·76-s − 632·79-s − 1.19e3·81-s + 3.68e3·89-s − 440·99-s + 2.26e3·101-s − 1.96e3·109-s − 6.91e3·116-s + 1.21e3·121-s − 1.63e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3·4-s + 0.370·9-s − 1.20·11-s + 4.93·16-s − 1.73·19-s − 1.84·29-s − 0.393·31-s + 10/9·36-s + 4.08·41-s − 3.61·44-s + 3.13·49-s − 2.79·59-s + 3.52·61-s + 45/8·64-s − 2.26·71-s − 5.21·76-s − 0.900·79-s − 1.63·81-s + 4.38·89-s − 0.446·99-s + 2.23·101-s − 1.72·109-s − 5.53·116-s + 0.909·121-s − 1.18·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.697096480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.697096480\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 p^{3} T^{2} + 65 p^{2} T^{4} - 3 p^{9} T^{6} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 1291 T^{4} - 10 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1076 T^{2} + 505542 T^{4} - 1076 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3188 T^{2} + 4514454 T^{4} - 3188 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 11100 T^{2} + 72435206 T^{4} - 11100 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 72 T + 4214 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36090 T^{2} + 584353451 T^{4} - 36090 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 144 T + 44554 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 34 T + 57519 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 200770 T^{2} + 15208041171 T^{4} - 200770 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 536 T + 209618 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 314692 T^{2} + 37397725014 T^{4} - 314692 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 291900 T^{2} + 39663724358 T^{4} - 291900 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 455660 T^{2} + 93957893046 T^{4} - 455660 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 634 T + 458975 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 840 T + 528794 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 916394 T^{2} + 390179035947 T^{4} - 916394 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 678 T + 813415 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 161924 T^{2} + 204091778022 T^{4} - 161924 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 p T - 279966 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2091236 T^{2} + 1737735911862 T^{4} - 2091236 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 1842 T + 1935427 T^{2} - 1842 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1119458 T^{2} + 1679793895011 T^{4} - 1119458 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81772965597152635372328368448, −7.78724059047881095123096835639, −7.69446751185059908294898000242, −7.42288124505880634011388211004, −7.15713222415703252182524346391, −6.89592836435243531860983383236, −6.78465561241106532973337912711, −6.17606190625744771821137522576, −6.13244279567184014590871469775, −5.93541461466281903632609737641, −5.71474725150857950411829879500, −5.35479056437784461504669182192, −5.10930925873633380741585036787, −4.40532914784115481577302433263, −4.20654532501381803700741064982, −4.05790502153425892943474322216, −3.52235338251476367130621421762, −3.11068212365901905711739257995, −2.64246023915408329191107017449, −2.56289422630260238368321333213, −2.25284509230590881466768617636, −1.80197377956965645335416910917, −1.74258435986295856006686620945, −0.851229721931016078657904539697, −0.50684220798334231562226717325,
0.50684220798334231562226717325, 0.851229721931016078657904539697, 1.74258435986295856006686620945, 1.80197377956965645335416910917, 2.25284509230590881466768617636, 2.56289422630260238368321333213, 2.64246023915408329191107017449, 3.11068212365901905711739257995, 3.52235338251476367130621421762, 4.05790502153425892943474322216, 4.20654532501381803700741064982, 4.40532914784115481577302433263, 5.10930925873633380741585036787, 5.35479056437784461504669182192, 5.71474725150857950411829879500, 5.93541461466281903632609737641, 6.13244279567184014590871469775, 6.17606190625744771821137522576, 6.78465561241106532973337912711, 6.89592836435243531860983383236, 7.15713222415703252182524346391, 7.42288124505880634011388211004, 7.69446751185059908294898000242, 7.78724059047881095123096835639, 7.81772965597152635372328368448
