| L(s) = 1 | − 2·2-s − 2·3-s + 4·4-s + 4·6-s − 21·7-s − 8·8-s − 23·9-s + 47·11-s − 8·12-s − 57·13-s + 42·14-s + 16·16-s + 84·17-s + 46·18-s − 5·19-s + 42·21-s − 94·22-s − 23·23-s + 16·24-s + 114·26-s + 100·27-s − 84·28-s + 285·29-s + 82·31-s − 32·32-s − 94·33-s − 168·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s + 1/2·4-s + 0.272·6-s − 1.13·7-s − 0.353·8-s − 0.851·9-s + 1.28·11-s − 0.192·12-s − 1.21·13-s + 0.801·14-s + 1/4·16-s + 1.19·17-s + 0.602·18-s − 0.0603·19-s + 0.436·21-s − 0.910·22-s − 0.208·23-s + 0.136·24-s + 0.859·26-s + 0.712·27-s − 0.566·28-s + 1.82·29-s + 0.475·31-s − 0.176·32-s − 0.495·33-s − 0.847·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 47 T + p^{3} T^{2} \) |
| 13 | \( 1 + 57 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 5 T + p^{3} T^{2} \) |
| 29 | \( 1 - 285 T + p^{3} T^{2} \) |
| 31 | \( 1 - 82 T + p^{3} T^{2} \) |
| 37 | \( 1 - 54 T + p^{3} T^{2} \) |
| 41 | \( 1 + 53 T + p^{3} T^{2} \) |
| 43 | \( 1 + 197 T + p^{3} T^{2} \) |
| 47 | \( 1 - 124 T + p^{3} T^{2} \) |
| 53 | \( 1 - 148 T + p^{3} T^{2} \) |
| 59 | \( 1 - 30 T + p^{3} T^{2} \) |
| 61 | \( 1 + 578 T + p^{3} T^{2} \) |
| 67 | \( 1 + 296 T + p^{3} T^{2} \) |
| 71 | \( 1 - 422 T + p^{3} T^{2} \) |
| 73 | \( 1 + 487 T + p^{3} T^{2} \) |
| 79 | \( 1 + 405 T + p^{3} T^{2} \) |
| 83 | \( 1 + 397 T + p^{3} T^{2} \) |
| 89 | \( 1 - 730 T + p^{3} T^{2} \) |
| 97 | \( 1 - 64 T + p^{3} T^{2} \) |
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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
−9.109701747319716660547024268323, −8.326908031894459659842895401909, −7.30186824685254416048561404802, −6.49036960445616779888171956341, −5.94397549809074339291102568156, −4.77336812364551196704007455815, −3.41971153013429439621171924927, −2.63165863094868126011073891523, −1.07327104076809931027681407775, 0,
1.07327104076809931027681407775, 2.63165863094868126011073891523, 3.41971153013429439621171924927, 4.77336812364551196704007455815, 5.94397549809074339291102568156, 6.49036960445616779888171956341, 7.30186824685254416048561404802, 8.326908031894459659842895401909, 9.109701747319716660547024268323
