L(s) = 1 | − 2.71·2-s + 1.57·3-s + 5.37·4-s + 5-s − 4.26·6-s + 1.27·7-s − 9.15·8-s − 0.531·9-s − 2.71·10-s + 11-s + 8.43·12-s + 1.99·13-s − 3.47·14-s + 1.57·15-s + 14.1·16-s − 4.79·17-s + 1.44·18-s − 0.355·19-s + 5.37·20-s + 2.00·21-s − 2.71·22-s + 1.37·23-s − 14.3·24-s + 25-s − 5.40·26-s − 5.54·27-s + 6.87·28-s + ⋯ |
L(s) = 1 | − 1.91·2-s + 0.907·3-s + 2.68·4-s + 0.447·5-s − 1.74·6-s + 0.483·7-s − 3.23·8-s − 0.177·9-s − 0.858·10-s + 0.301·11-s + 2.43·12-s + 0.552·13-s − 0.928·14-s + 0.405·15-s + 3.52·16-s − 1.16·17-s + 0.340·18-s − 0.0816·19-s + 1.20·20-s + 0.438·21-s − 0.578·22-s + 0.287·23-s − 2.93·24-s + 0.200·25-s − 1.06·26-s − 1.06·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 3 | \( 1 - 1.57T + 3T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 13 | \( 1 - 1.99T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 + 0.355T + 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 - 1.99T + 37T^{2} \) |
| 41 | \( 1 + 5.40T + 41T^{2} \) |
| 43 | \( 1 + 5.32T + 43T^{2} \) |
| 47 | \( 1 - 8.84T + 47T^{2} \) |
| 53 | \( 1 + 7.05T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 8.88T + 67T^{2} \) |
| 71 | \( 1 + 8.65T + 71T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 6.54T + 83T^{2} \) |
| 89 | \( 1 - 0.603T + 89T^{2} \) |
| 97 | \( 1 + 2.67T + 97T^{2} \) |
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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
−8.481801538878562520553309515464, −7.57406462935181192988029021797, −7.02403587128247696597215148409, −6.19327104911231780019891983803, −5.41729196491841690040502753804, −3.88567481688647077948763312627, −2.94915740785686815555861255643, −2.04731694152009267678160814725, −1.54992335423290495248639718259, 0,
1.54992335423290495248639718259, 2.04731694152009267678160814725, 2.94915740785686815555861255643, 3.88567481688647077948763312627, 5.41729196491841690040502753804, 6.19327104911231780019891983803, 7.02403587128247696597215148409, 7.57406462935181192988029021797, 8.481801538878562520553309515464