| L(s) = 1 | − 2·2-s − 5·3-s + 4·4-s − 5·5-s + 10·6-s + 12·7-s − 8·8-s − 2·9-s + 10·10-s + 22·11-s − 20·12-s + 19·13-s − 24·14-s + 25·15-s + 16·16-s + 96·17-s + 4·18-s − 98·19-s − 20·20-s − 60·21-s − 44·22-s + 23·23-s + 40·24-s + 25·25-s − 38·26-s + 145·27-s + 48·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.962·3-s + 1/2·4-s − 0.447·5-s + 0.680·6-s + 0.647·7-s − 0.353·8-s − 0.0740·9-s + 0.316·10-s + 0.603·11-s − 0.481·12-s + 0.405·13-s − 0.458·14-s + 0.430·15-s + 1/4·16-s + 1.36·17-s + 0.0523·18-s − 1.18·19-s − 0.223·20-s − 0.623·21-s − 0.426·22-s + 0.208·23-s + 0.340·24-s + 1/5·25-s − 0.286·26-s + 1.03·27-s + 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 17 | \( 1 - 96 T + p^{3} T^{2} \) |
| 19 | \( 1 + 98 T + p^{3} T^{2} \) |
| 29 | \( 1 + 227 T + p^{3} T^{2} \) |
| 31 | \( 1 + 285 T + p^{3} T^{2} \) |
| 37 | \( 1 + 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 271 T + p^{3} T^{2} \) |
| 43 | \( 1 + 100 T + p^{3} T^{2} \) |
| 47 | \( 1 + 285 T + p^{3} T^{2} \) |
| 53 | \( 1 - 18 T + p^{3} T^{2} \) |
| 59 | \( 1 + 352 T + p^{3} T^{2} \) |
| 61 | \( 1 + 478 T + p^{3} T^{2} \) |
| 67 | \( 1 - 330 T + p^{3} T^{2} \) |
| 71 | \( 1 - 835 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1127 T + p^{3} T^{2} \) |
| 79 | \( 1 - 322 T + p^{3} T^{2} \) |
| 83 | \( 1 - 572 T + p^{3} T^{2} \) |
| 89 | \( 1 + 504 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1712 T + p^{3} T^{2} \) |
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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
−11.13979251595794782717905242440, −10.63460978918105054249021931931, −9.271146026982274550913964704526, −8.314517375809311569584384074864, −7.30975622896575183556549311400, −6.16357963915471120031940393614, −5.17303004103564603350685706206, −3.62119645496589894030215795952, −1.54999023609572014001308635739, 0,
1.54999023609572014001308635739, 3.62119645496589894030215795952, 5.17303004103564603350685706206, 6.16357963915471120031940393614, 7.30975622896575183556549311400, 8.314517375809311569584384074864, 9.271146026982274550913964704526, 10.63460978918105054249021931931, 11.13979251595794782717905242440
