| L(s) = 1 | − 3·4-s − 3·5-s − 2·9-s − 10·11-s + 5·16-s − 8·19-s + 9·20-s + 4·25-s − 12·29-s + 6·36-s + 10·41-s + 30·44-s + 6·45-s − 13·49-s + 30·55-s + 18·59-s − 2·61-s − 3·64-s − 16·71-s + 24·76-s + 6·79-s − 15·80-s − 5·81-s − 8·89-s + 24·95-s + 20·99-s − 12·100-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.34·5-s − 2/3·9-s − 3.01·11-s + 5/4·16-s − 1.83·19-s + 2.01·20-s + 4/5·25-s − 2.22·29-s + 36-s + 1.56·41-s + 4.52·44-s + 0.894·45-s − 1.85·49-s + 4.04·55-s + 2.34·59-s − 0.256·61-s − 3/8·64-s − 1.89·71-s + 2.75·76-s + 0.675·79-s − 1.67·80-s − 5/9·81-s − 0.847·89-s + 2.46·95-s + 2.01·99-s − 6/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.019416501615119066110839147797, −8.400937388344254657063163947760, −8.287570693733701613432166498970, −7.66039763333771123227596439320, −7.58590694377702174449965560919, −6.61981736464800576569297638001, −5.60157612088078313975836309422, −5.50181452599748063455305172341, −4.82603389209875681234034335898, −4.25666915587159511841100490901, −3.82229793578931859480218288321, −2.97340716320340968167510315654, −2.30723040854003748390037888337, 0, 0,
2.30723040854003748390037888337, 2.97340716320340968167510315654, 3.82229793578931859480218288321, 4.25666915587159511841100490901, 4.82603389209875681234034335898, 5.50181452599748063455305172341, 5.60157612088078313975836309422, 6.61981736464800576569297638001, 7.58590694377702174449965560919, 7.66039763333771123227596439320, 8.287570693733701613432166498970, 8.400937388344254657063163947760, 9.019416501615119066110839147797
