| L(s) = 1 | + 2-s + 4-s + 4·5-s + 7-s + 8-s + 4·10-s − 2·11-s − 2·13-s + 14-s + 16-s + 19-s + 4·20-s − 2·22-s − 2·23-s + 11·25-s − 2·26-s + 28-s + 6·29-s + 4·31-s + 32-s + 4·35-s + 2·37-s + 38-s + 4·40-s + 10·41-s + 4·43-s − 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s + 0.353·8-s + 1.26·10-s − 0.603·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.229·19-s + 0.894·20-s − 0.426·22-s − 0.417·23-s + 11/5·25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.676·35-s + 0.328·37-s + 0.162·38-s + 0.632·40-s + 1.56·41-s + 0.609·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.069382754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.069382754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.113944302689522844954497345768, −8.128370287098792480676945535600, −7.28401581826763035105333257221, −6.32634043889203686982122190390, −5.83860720256681699532957618907, −5.05299406609889339629338369738, −4.42887591677329679176409186185, −2.90145174599352706260615069014, −2.36353698091510760667675177945, −1.31158692776558981751888895743,
1.31158692776558981751888895743, 2.36353698091510760667675177945, 2.90145174599352706260615069014, 4.42887591677329679176409186185, 5.05299406609889339629338369738, 5.83860720256681699532957618907, 6.32634043889203686982122190390, 7.28401581826763035105333257221, 8.128370287098792480676945535600, 9.113944302689522844954497345768
