| L(s) = 1 | + 2.51·2-s − 1.51·3-s + 4.32·4-s − 3.80·6-s + 3.32·7-s + 5.83·8-s − 0.707·9-s + 2.83·11-s − 6.54·12-s + 8.34·14-s + 6.02·16-s + 6.64·17-s − 1.77·18-s − 2.19·19-s − 5.02·21-s + 7.12·22-s + 0.485·23-s − 8.83·24-s + 5.61·27-s + 14.3·28-s − 3.32·29-s − 3.80·31-s + 3.48·32-s − 4.29·33-s + 16.6·34-s − 3.05·36-s + 9.32·37-s + ⋯ |
| L(s) = 1 | + 1.77·2-s − 0.874·3-s + 2.16·4-s − 1.55·6-s + 1.25·7-s + 2.06·8-s − 0.235·9-s + 0.854·11-s − 1.88·12-s + 2.23·14-s + 1.50·16-s + 1.61·17-s − 0.419·18-s − 0.503·19-s − 1.09·21-s + 1.51·22-s + 0.101·23-s − 1.80·24-s + 1.08·27-s + 2.71·28-s − 0.616·29-s − 0.683·31-s + 0.616·32-s − 0.747·33-s + 2.86·34-s − 0.509·36-s + 1.53·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.417918237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.417918237\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 11 | \( 1 - 2.83T + 11T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 - 0.485T + 23T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 31 | \( 1 + 3.80T + 31T^{2} \) |
| 37 | \( 1 - 9.32T + 37T^{2} \) |
| 41 | \( 1 - 1.61T + 41T^{2} \) |
| 43 | \( 1 - 0.872T + 43T^{2} \) |
| 47 | \( 1 + 3.32T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.83T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 4.29T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 0.585T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 - 0.641T + 97T^{2} \) |
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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
−8.030121159898190565697358275125, −7.47049692908575460610623043092, −6.46646187360900324832366023213, −5.99906292800262090793532829992, −5.28010622432883991232158052446, −4.84023708600086658924501290271, −4.01840767942760136811017778846, −3.26090964565783744623141799728, −2.14024449977899925222368727984, −1.15307989065850026103309113851,
1.15307989065850026103309113851, 2.14024449977899925222368727984, 3.26090964565783744623141799728, 4.01840767942760136811017778846, 4.84023708600086658924501290271, 5.28010622432883991232158052446, 5.99906292800262090793532829992, 6.46646187360900324832366023213, 7.47049692908575460610623043092, 8.030121159898190565697358275125
