| L(s) = 1 | + 0.737·2-s − 1.37·3-s − 1.45·4-s − 5-s − 1.01·6-s + 7-s − 2.54·8-s − 1.10·9-s − 0.737·10-s + 2.00·12-s + 2.32·13-s + 0.737·14-s + 1.37·15-s + 1.03·16-s + 7.63·17-s − 0.813·18-s − 5.70·19-s + 1.45·20-s − 1.37·21-s − 2.54·23-s + 3.51·24-s + 25-s + 1.71·26-s + 5.65·27-s − 1.45·28-s − 10.5·29-s + 1.01·30-s + ⋯ |
| L(s) = 1 | + 0.521·2-s − 0.795·3-s − 0.727·4-s − 0.447·5-s − 0.414·6-s + 0.377·7-s − 0.901·8-s − 0.367·9-s − 0.233·10-s + 0.578·12-s + 0.645·13-s + 0.197·14-s + 0.355·15-s + 0.257·16-s + 1.85·17-s − 0.191·18-s − 1.30·19-s + 0.325·20-s − 0.300·21-s − 0.530·23-s + 0.716·24-s + 0.200·25-s + 0.336·26-s + 1.08·27-s − 0.275·28-s − 1.95·29-s + 0.185·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.737T + 2T^{2} \) |
| 3 | \( 1 + 1.37T + 3T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 - 7.63T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 0.781T + 31T^{2} \) |
| 37 | \( 1 - 0.922T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 + 7.79T + 61T^{2} \) |
| 67 | \( 1 + 3.88T + 67T^{2} \) |
| 71 | \( 1 + 4.91T + 71T^{2} \) |
| 73 | \( 1 + 2.46T + 73T^{2} \) |
| 79 | \( 1 - 4.15T + 79T^{2} \) |
| 83 | \( 1 + 8.52T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 - 2.30T + 97T^{2} \) |
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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
−7.968740177785948247783864479185, −7.40310609114494750187759469727, −6.08224303867333674958083525966, −5.80285810023086829722985810829, −5.16917048762176378712032454032, −4.14960681385352164034058807247, −3.76976124263982166013454019187, −2.65226131775523872930993710200, −1.13009444966275809242174951565, 0,
1.13009444966275809242174951565, 2.65226131775523872930993710200, 3.76976124263982166013454019187, 4.14960681385352164034058807247, 5.16917048762176378712032454032, 5.80285810023086829722985810829, 6.08224303867333674958083525966, 7.40310609114494750187759469727, 7.968740177785948247783864479185
