L(s) = 1 | + 2-s + 1.53·3-s + 4-s + 0.384·5-s + 1.53·6-s − 3.26·7-s + 8-s − 0.652·9-s + 0.384·10-s − 2.58·11-s + 1.53·12-s + 1.36·13-s − 3.26·14-s + 0.589·15-s + 16-s − 3.33·17-s − 0.652·18-s − 5.98·19-s + 0.384·20-s − 5.00·21-s − 2.58·22-s − 7.54·23-s + 1.53·24-s − 4.85·25-s + 1.36·26-s − 5.59·27-s − 3.26·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.884·3-s + 0.5·4-s + 0.172·5-s + 0.625·6-s − 1.23·7-s + 0.353·8-s − 0.217·9-s + 0.121·10-s − 0.779·11-s + 0.442·12-s + 0.378·13-s − 0.872·14-s + 0.152·15-s + 0.250·16-s − 0.808·17-s − 0.153·18-s − 1.37·19-s + 0.0860·20-s − 1.09·21-s − 0.551·22-s − 1.57·23-s + 0.312·24-s − 0.970·25-s + 0.267·26-s − 1.07·27-s − 0.616·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 5 | \( 1 - 0.384T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + 5.98T + 19T^{2} \) |
| 23 | \( 1 + 7.54T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 - 4.33T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 + 8.67T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 7.01T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 + 0.0111T + 83T^{2} \) |
| 89 | \( 1 - 0.953T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.265886828044909332230281169226, −7.86613947857066995120947979022, −6.70010078401219262811408067901, −6.14778489889865176856525482301, −5.46842982116854896479763930621, −4.10734772180724244947924132457, −3.72403399078832705505722226661, −2.55187289810528314235806417316, −2.20726955691363224680644303296, 0,
2.20726955691363224680644303296, 2.55187289810528314235806417316, 3.72403399078832705505722226661, 4.10734772180724244947924132457, 5.46842982116854896479763930621, 6.14778489889865176856525482301, 6.70010078401219262811408067901, 7.86613947857066995120947979022, 8.265886828044909332230281169226