| L(s) = 1 | − 2-s + 4·3-s − 7·4-s − 4·6-s + 7·7-s + 15·8-s − 11·9-s − 12·11-s − 28·12-s − 13·13-s − 7·14-s + 41·16-s + 94·17-s + 11·18-s − 4·19-s + 28·21-s + 12·22-s + 24·23-s + 60·24-s + 13·26-s − 152·27-s − 49·28-s + 6·29-s − 160·31-s − 161·32-s − 48·33-s − 94·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.769·3-s − 7/8·4-s − 0.272·6-s + 0.377·7-s + 0.662·8-s − 0.407·9-s − 0.328·11-s − 0.673·12-s − 0.277·13-s − 0.133·14-s + 0.640·16-s + 1.34·17-s + 0.144·18-s − 0.0482·19-s + 0.290·21-s + 0.116·22-s + 0.217·23-s + 0.510·24-s + 0.0980·26-s − 1.08·27-s − 0.330·28-s + 0.0384·29-s − 0.926·31-s − 0.889·32-s − 0.253·33-s − 0.474·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 24 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 350 T + p^{3} T^{2} \) |
| 41 | \( 1 - 362 T + p^{3} T^{2} \) |
| 43 | \( 1 + 452 T + p^{3} T^{2} \) |
| 47 | \( 1 - 384 T + p^{3} T^{2} \) |
| 53 | \( 1 - 546 T + p^{3} T^{2} \) |
| 59 | \( 1 - 148 T + p^{3} T^{2} \) |
| 61 | \( 1 - 870 T + p^{3} T^{2} \) |
| 67 | \( 1 + 396 T + p^{3} T^{2} \) |
| 71 | \( 1 - 744 T + p^{3} T^{2} \) |
| 73 | \( 1 + 474 T + p^{3} T^{2} \) |
| 79 | \( 1 + 176 T + p^{3} T^{2} \) |
| 83 | \( 1 + 188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 646 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1090 T + p^{3} T^{2} \) |
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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
−8.429598568745299356825173702233, −7.74129279445281495902451197225, −7.09976886742106601329973109896, −5.57310477107925412891858801051, −5.26400804996288723278777499222, −4.05675046871227239836856389455, −3.37015227986892586422001678165, −2.32029695808599390710346085496, −1.17531691321505049663116250937, 0,
1.17531691321505049663116250937, 2.32029695808599390710346085496, 3.37015227986892586422001678165, 4.05675046871227239836856389455, 5.26400804996288723278777499222, 5.57310477107925412891858801051, 7.09976886742106601329973109896, 7.74129279445281495902451197225, 8.429598568745299356825173702233
