| L(s) = 1 | − 2.52e4·13-s + 1.56e5·25-s − 1.23e6·37-s + 1.46e6·49-s + 3.07e6·61-s − 2.47e6·73-s − 1.05e7·97-s + 3.36e7·109-s − 3.89e7·121-s + ⋯ |
| L(s) = 1 | − 3.18·13-s + 2·25-s − 3.99·37-s + 1.77·49-s + 1.73·61-s − 0.744·73-s − 1.17·97-s + 2.48·109-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6936793497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6936793497\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 1763 T + p^{7} T^{2} )( 1 + 1763 T + p^{7} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 12605 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 14357 T + p^{7} T^{2} )( 1 + 14357 T + p^{7} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 178916 T + p^{7} T^{2} )( 1 + 178916 T + p^{7} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 615373 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1035224 T + p^{7} T^{2} )( 1 + 1035224 T + p^{7} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1537199 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4058455 T + p^{7} T^{2} )( 1 + 4058455 T + p^{7} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1236809 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4245427 T + p^{7} T^{2} )( 1 + 4245427 T + p^{7} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5276357 T + p^{7} T^{2} )^{2} \) |
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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
−10.28124544474391650507963097385, −9.854096409579295704220333260017, −9.291273334620004486279540054551, −8.870845889099850765672517791439, −8.498729347685930650202221568749, −7.903952817096794424208509517347, −7.08596360382006687698748375659, −7.07038587813233116515961678571, −6.94844110266371364265581663065, −5.89715742325800446623030582450, −5.21979951294364707358277709395, −5.07370450848365998667228193723, −4.66532975297591520111734033315, −3.91745555894999348523358904367, −3.25840356030531079304767701302, −2.69112808039643191097324308790, −2.27504938584138911590881437986, −1.70595149484928790739803382386, −0.851270297659182734892873284588, −0.19697802685346148365508166064,
0.19697802685346148365508166064, 0.851270297659182734892873284588, 1.70595149484928790739803382386, 2.27504938584138911590881437986, 2.69112808039643191097324308790, 3.25840356030531079304767701302, 3.91745555894999348523358904367, 4.66532975297591520111734033315, 5.07370450848365998667228193723, 5.21979951294364707358277709395, 5.89715742325800446623030582450, 6.94844110266371364265581663065, 7.07038587813233116515961678571, 7.08596360382006687698748375659, 7.903952817096794424208509517347, 8.498729347685930650202221568749, 8.870845889099850765672517791439, 9.291273334620004486279540054551, 9.854096409579295704220333260017, 10.28124544474391650507963097385
