| L(s) = 1 | − 3·2-s + 94·3-s − 33·4-s + 330·5-s − 282·6-s − 686·7-s − 159·8-s + 3.11e3·9-s − 990·10-s + 2.84e3·11-s − 3.10e3·12-s + 2.53e3·13-s + 2.05e3·14-s + 3.10e4·15-s − 1.33e4·16-s − 1.48e3·17-s − 9.35e3·18-s + 3.28e4·19-s − 1.08e4·20-s − 6.44e4·21-s − 8.53e3·22-s − 6.57e3·23-s − 1.49e4·24-s − 5.29e4·25-s − 7.60e3·26-s − 3.88e4·27-s + 2.26e4·28-s + ⋯ |
| L(s) = 1 | − 0.265·2-s + 2.01·3-s − 0.257·4-s + 1.18·5-s − 0.532·6-s − 0.755·7-s − 0.109·8-s + 1.42·9-s − 0.313·10-s + 0.644·11-s − 0.518·12-s + 0.319·13-s + 0.200·14-s + 2.37·15-s − 0.815·16-s − 0.0734·17-s − 0.378·18-s + 1.09·19-s − 0.304·20-s − 1.51·21-s − 0.170·22-s − 0.112·23-s − 0.220·24-s − 0.677·25-s − 0.0848·26-s − 0.379·27-s + 0.194·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.349267935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.349267935\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 3 T + 21 p T^{2} + 3 p^{7} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 94 T + 1906 p T^{2} - 94 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 66 p T + 6474 p^{2} T^{2} - 66 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2844 T + 38086566 T^{2} - 2844 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2534 T - 41123742 T^{2} - 2534 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1488 T + 798529822 T^{2} + 1488 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32810 T + 1897672038 T^{2} - 32810 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6576 T + 6819963598 T^{2} + 6576 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 20640 T + 15579628518 T^{2} - 20640 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 391836 T + 92048864606 T^{2} + 391836 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 367392 T + 63852768182 T^{2} - 367392 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 734664 T + 402811824126 T^{2} - 734664 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 480476 T + 594501933318 T^{2} + 480476 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1089108 T + 1015337137342 T^{2} + 1089108 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2858844 T + 4386858062398 T^{2} - 2858844 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 160170 T + 4361928868198 T^{2} - 160170 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 864646 T + 5755969170906 T^{2} + 864646 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 328648 T + 11587546356582 T^{2} + 328648 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7500216 T + 28549732695406 T^{2} + 7500216 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4301244 T + 18754109784038 T^{2} - 4301244 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6408440 T + 32072611946718 T^{2} + 6408440 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11659074 T + 84453675852838 T^{2} - 11659074 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9772260 T + 83812995056598 T^{2} - 9772260 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10762752 T + 188617737573662 T^{2} - 10762752 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.14357199722605128099117789009, −20.33405820931761331150310396742, −19.85703242294188046364948049321, −19.42753266438982118507604672467, −18.21065847111089065376569316290, −17.99615491591178053259269306805, −16.73539889738544817238284461214, −15.92886992242629341607193624881, −14.75283466028508829624793004813, −14.33278475838902338493712788027, −13.35729929638836271104508273974, −13.31947393752397283791288230075, −11.59040495312761125574552358148, −9.938860129374097454142208376625, −9.216437502843415366742090070228, −8.930689947234920834672850754946, −7.54844764954608328007123844444, −5.98912370532902750246157725918, −3.54736914293094747875810162021, −2.21275473409769419015641267435,
2.21275473409769419015641267435, 3.54736914293094747875810162021, 5.98912370532902750246157725918, 7.54844764954608328007123844444, 8.930689947234920834672850754946, 9.216437502843415366742090070228, 9.938860129374097454142208376625, 11.59040495312761125574552358148, 13.31947393752397283791288230075, 13.35729929638836271104508273974, 14.33278475838902338493712788027, 14.75283466028508829624793004813, 15.92886992242629341607193624881, 16.73539889738544817238284461214, 17.99615491591178053259269306805, 18.21065847111089065376569316290, 19.42753266438982118507604672467, 19.85703242294188046364948049321, 20.33405820931761331150310396742, 21.14357199722605128099117789009
