| L(s) = 1 | + 8·2-s + 43·3-s + 64·4-s + 344·6-s + 974·7-s + 512·8-s − 338·9-s + 87·11-s + 2.75e3·12-s + 1.48e4·13-s + 7.79e3·14-s + 4.09e3·16-s − 3.55e4·17-s − 2.70e3·18-s + 2.06e4·19-s + 4.18e4·21-s + 696·22-s + 2.22e4·23-s + 2.20e4·24-s + 1.18e5·26-s − 1.08e5·27-s + 6.23e4·28-s − 5.76e3·29-s + 3.02e5·31-s + 3.27e4·32-s + 3.74e3·33-s − 2.84e5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.919·3-s + 1/2·4-s + 0.650·6-s + 1.07·7-s + 0.353·8-s − 0.154·9-s + 0.0197·11-s + 0.459·12-s + 1.87·13-s + 0.758·14-s + 1/4·16-s − 1.75·17-s − 0.109·18-s + 0.689·19-s + 0.986·21-s + 0.0139·22-s + 0.380·23-s + 0.325·24-s + 1.32·26-s − 1.06·27-s + 0.536·28-s − 0.0438·29-s + 1.82·31-s + 0.176·32-s + 0.0181·33-s − 1.24·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.143434192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.143434192\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 43 T + p^{7} T^{2} \) |
| 7 | \( 1 - 974 T + p^{7} T^{2} \) |
| 11 | \( 1 - 87 T + p^{7} T^{2} \) |
| 13 | \( 1 - 14828 T + p^{7} T^{2} \) |
| 17 | \( 1 + 35571 T + p^{7} T^{2} \) |
| 19 | \( 1 - 1085 p T + p^{7} T^{2} \) |
| 23 | \( 1 - 42 p^{2} T + p^{7} T^{2} \) |
| 29 | \( 1 + 5760 T + p^{7} T^{2} \) |
| 31 | \( 1 - 302942 T + p^{7} T^{2} \) |
| 37 | \( 1 + 199366 T + p^{7} T^{2} \) |
| 41 | \( 1 + 668523 T + p^{7} T^{2} \) |
| 43 | \( 1 + 143212 T + p^{7} T^{2} \) |
| 47 | \( 1 + 338316 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1094322 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2135520 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1939318 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3348751 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3005652 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3048397 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5485130 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5205933 T + p^{7} T^{2} \) |
| 89 | \( 1 + 832665 T + p^{7} T^{2} \) |
| 97 | \( 1 - 5314754 T + p^{7} T^{2} \) |
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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
−13.79414947288227901931867372671, −13.50239394834035152784739323064, −11.67918209715334951580709388365, −10.85345983737617165694505311437, −8.888174722354831899102768042706, −8.065553362880433584233309204490, −6.36491566534283492412585771464, −4.69574392619821651095452801115, −3.25012261275117587783173566296, −1.67875777347784413776077382503,
1.67875777347784413776077382503, 3.25012261275117587783173566296, 4.69574392619821651095452801115, 6.36491566534283492412585771464, 8.065553362880433584233309204490, 8.888174722354831899102768042706, 10.85345983737617165694505311437, 11.67918209715334951580709388365, 13.50239394834035152784739323064, 13.79414947288227901931867372671
