| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 4·11-s + 12-s − 13-s − 16-s + 18-s − 4·19-s − 4·22-s + 3·24-s − 26-s − 27-s + 2·29-s + 8·31-s + 5·32-s + 4·33-s − 36-s − 2·37-s − 4·38-s + 39-s − 2·41-s + 4·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.852·22-s + 0.612·24-s − 0.196·26-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.883·32-s + 0.696·33-s − 1/6·36-s − 0.328·37-s − 0.648·38-s + 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.90525475870083, −12.68094906675646, −12.06691623740783, −11.80068548259619, −11.22089034683833, −10.64107087773913, −10.24762815765257, −9.916132497699575, −9.338162433219187, −8.748568364254669, −8.278666456880743, −7.953833254305433, −7.269594898559256, −6.719958539950215, −6.180901650064265, −5.861990218869029, −5.238478082586779, −4.787283133804468, −4.565209768235303, −3.927451457029654, −3.288752540619294, −2.748130570902686, −2.272450961559735, −1.409919667509117, −0.5681149720156749, 0,
0.5681149720156749, 1.409919667509117, 2.272450961559735, 2.748130570902686, 3.288752540619294, 3.927451457029654, 4.565209768235303, 4.787283133804468, 5.238478082586779, 5.861990218869029, 6.180901650064265, 6.719958539950215, 7.269594898559256, 7.953833254305433, 8.278666456880743, 8.748568364254669, 9.338162433219187, 9.916132497699575, 10.24762815765257, 10.64107087773913, 11.22089034683833, 11.80068548259619, 12.06691623740783, 12.68094906675646, 12.90525475870083
