| L(s) = 1 | + 2.44·2-s + 3.99·4-s − 5-s + 4.85·7-s + 4.89·8-s − 2.44·10-s + 6.48·11-s − 3.08·13-s + 11.8·14-s + 3.99·16-s − 2.06·17-s + 3.08·19-s − 3.99·20-s + 15.8·22-s + 0.433·23-s + 25-s − 7.56·26-s + 19.4·28-s − 8.86·29-s + 10.6·31-s − 0.0151·32-s − 5.05·34-s − 4.85·35-s − 7.75·37-s + 7.56·38-s − 4.89·40-s + 2.82·41-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 1.99·4-s − 0.447·5-s + 1.83·7-s + 1.73·8-s − 0.774·10-s + 1.95·11-s − 0.856·13-s + 3.17·14-s + 0.997·16-s − 0.500·17-s + 0.708·19-s − 0.894·20-s + 3.38·22-s + 0.0903·23-s + 0.200·25-s − 1.48·26-s + 3.66·28-s − 1.64·29-s + 1.91·31-s − 0.00267·32-s − 0.866·34-s − 0.820·35-s − 1.27·37-s + 1.22·38-s − 0.773·40-s + 0.441·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.925925992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.925925992\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 7 | \( 1 - 4.85T + 7T^{2} \) |
| 11 | \( 1 - 6.48T + 11T^{2} \) |
| 13 | \( 1 + 3.08T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 - 0.433T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 + 4.99T + 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 + 9.32T + 53T^{2} \) |
| 59 | \( 1 - 4.40T + 59T^{2} \) |
| 61 | \( 1 - 9.44T + 61T^{2} \) |
| 67 | \( 1 + 0.197T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 1.68T + 73T^{2} \) |
| 79 | \( 1 - 5.54T + 79T^{2} \) |
| 83 | \( 1 + 2.81T + 83T^{2} \) |
| 97 | \( 1 + 6.95T + 97T^{2} \) |
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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
−8.275014807959744388525167361869, −7.38805946156358964386482450712, −6.90209508948134471086576117275, −6.05551266411923554574657981525, −5.13823843722920507162650463741, −4.65716469975457543359981540984, −4.08015454671946326556540621910, −3.30569653099217295707906182251, −2.11582142338449808243424394004, −1.37429709636709085135492848410,
1.37429709636709085135492848410, 2.11582142338449808243424394004, 3.30569653099217295707906182251, 4.08015454671946326556540621910, 4.65716469975457543359981540984, 5.13823843722920507162650463741, 6.05551266411923554574657981525, 6.90209508948134471086576117275, 7.38805946156358964386482450712, 8.275014807959744388525167361869
