| L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s + 11-s − 2·13-s + 16-s + 3·17-s − 2·19-s − 3·20-s + 22-s − 3·23-s + 4·25-s − 2·26-s + 6·29-s + 4·31-s + 32-s + 3·34-s + 2·37-s − 2·38-s − 3·40-s − 3·41-s + 2·43-s + 44-s − 3·46-s − 9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.727·17-s − 0.458·19-s − 0.670·20-s + 0.213·22-s − 0.625·23-s + 4/5·25-s − 0.392·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s − 0.324·38-s − 0.474·40-s − 0.468·41-s + 0.304·43-s + 0.150·44-s − 0.442·46-s − 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.36682342264922509928167997297, −6.56168671042350513543704200111, −6.09381729734312770986104002781, −4.95928831043882073933100303728, −4.61656435962284028919457906094, −3.80129494461301145830993283882, −3.26126018618869786139603167399, −2.40963137978737592075226396775, −1.23424179187820142157076819828, 0,
1.23424179187820142157076819828, 2.40963137978737592075226396775, 3.26126018618869786139603167399, 3.80129494461301145830993283882, 4.61656435962284028919457906094, 4.95928831043882073933100303728, 6.09381729734312770986104002781, 6.56168671042350513543704200111, 7.36682342264922509928167997297
