| L(s) = 1 | + 1.07·2-s − 3-s − 0.836·4-s − 2.04·5-s − 1.07·6-s − 3.13·7-s − 3.05·8-s + 9-s − 2.20·10-s + 0.618·11-s + 0.836·12-s + 1.06·13-s − 3.38·14-s + 2.04·15-s − 1.62·16-s + 17-s + 1.07·18-s + 0.790·19-s + 1.70·20-s + 3.13·21-s + 0.666·22-s + 3.33·23-s + 3.05·24-s − 0.836·25-s + 1.14·26-s − 27-s + 2.62·28-s + ⋯ |
| L(s) = 1 | + 0.762·2-s − 0.577·3-s − 0.418·4-s − 0.912·5-s − 0.440·6-s − 1.18·7-s − 1.08·8-s + 0.333·9-s − 0.695·10-s + 0.186·11-s + 0.241·12-s + 0.294·13-s − 0.904·14-s + 0.526·15-s − 0.406·16-s + 0.242·17-s + 0.254·18-s + 0.181·19-s + 0.381·20-s + 0.684·21-s + 0.142·22-s + 0.694·23-s + 0.624·24-s − 0.167·25-s + 0.224·26-s − 0.192·27-s + 0.496·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 1.07T + 2T^{2} \) |
| 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 19 | \( 1 - 0.790T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 + 0.155T + 29T^{2} \) |
| 31 | \( 1 + 5.97T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 - 0.422T + 47T^{2} \) |
| 53 | \( 1 + 0.695T + 53T^{2} \) |
| 59 | \( 1 - 3.59T + 59T^{2} \) |
| 61 | \( 1 + 2.59T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 3.87T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 - 4.21T + 83T^{2} \) |
| 89 | \( 1 - 0.820T + 89T^{2} \) |
| 97 | \( 1 - 9.25T + 97T^{2} \) |
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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
−7.41478394998027182419026252728, −6.54819337294282842852461996379, −6.06313921661171428898848168690, −5.36867294130039899507011232519, −4.61033183268132594786765138507, −3.82957238494235066298765314000, −3.50628190374303350579293733423, −2.58197002035366158728320820994, −0.919702366387890097539239512392, 0,
0.919702366387890097539239512392, 2.58197002035366158728320820994, 3.50628190374303350579293733423, 3.82957238494235066298765314000, 4.61033183268132594786765138507, 5.36867294130039899507011232519, 6.06313921661171428898848168690, 6.54819337294282842852461996379, 7.41478394998027182419026252728
