L(s) = 1 | − 2-s + 4-s − 3.31·5-s − 3.41·7-s − 8-s + 3.31·10-s − 3.88·11-s − 0.659·13-s + 3.41·14-s + 16-s − 1.70·17-s + 2.86·19-s − 3.31·20-s + 3.88·22-s − 3.18·23-s + 6.01·25-s + 0.659·26-s − 3.41·28-s − 1.15·29-s − 0.368·31-s − 32-s + 1.70·34-s + 11.3·35-s + 5.75·37-s − 2.86·38-s + 3.31·40-s + 6.68·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.48·5-s − 1.29·7-s − 0.353·8-s + 1.04·10-s − 1.17·11-s − 0.183·13-s + 0.913·14-s + 0.250·16-s − 0.413·17-s + 0.657·19-s − 0.742·20-s + 0.827·22-s − 0.663·23-s + 1.20·25-s + 0.129·26-s − 0.646·28-s − 0.214·29-s − 0.0661·31-s − 0.176·32-s + 0.292·34-s + 1.91·35-s + 0.946·37-s − 0.465·38-s + 0.524·40-s + 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 13 | \( 1 + 0.659T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 + 1.15T + 29T^{2} \) |
| 31 | \( 1 + 0.368T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 - 6.68T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 2.43T + 53T^{2} \) |
| 59 | \( 1 + 5.82T + 59T^{2} \) |
| 61 | \( 1 + 0.592T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 + 1.63T + 71T^{2} \) |
| 73 | \( 1 + 0.306T + 73T^{2} \) |
| 79 | \( 1 + 4.61T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 8.75T + 97T^{2} \) |
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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
−7.56380610280564140170896554818, −7.14674488179754468462087781412, −6.18587390103175919350592156431, −5.59571703996241273848949704568, −4.45584421511168035288431010164, −3.83530571857973710075498036168, −2.97444686512132501628578051251, −2.43591984141413704322074058579, −0.77325335231765937139230839026, 0,
0.77325335231765937139230839026, 2.43591984141413704322074058579, 2.97444686512132501628578051251, 3.83530571857973710075498036168, 4.45584421511168035288431010164, 5.59571703996241273848949704568, 6.18587390103175919350592156431, 7.14674488179754468462087781412, 7.56380610280564140170896554818