| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s − 3.41·5-s + 2.41·6-s + 2.82·7-s − 4.41·8-s + 9-s + 8.24·10-s + 1.41·11-s − 3.82·12-s + 13-s − 6.82·14-s + 3.41·15-s + 2.99·16-s + 1.17·17-s − 2.41·18-s + 5.65·19-s − 13.0·20-s − 2.82·21-s − 3.41·22-s − 3.65·23-s + 4.41·24-s + 6.65·25-s − 2.41·26-s − 27-s + 10.8·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s − 1.52·5-s + 0.985·6-s + 1.06·7-s − 1.56·8-s + 0.333·9-s + 2.60·10-s + 0.426·11-s − 1.10·12-s + 0.277·13-s − 1.82·14-s + 0.881·15-s + 0.749·16-s + 0.284·17-s − 0.569·18-s + 1.29·19-s − 2.92·20-s − 0.617·21-s − 0.727·22-s − 0.762·23-s + 0.901·24-s + 1.33·25-s − 0.473·26-s − 0.192·27-s + 2.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 7.41T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 4.82T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.87073897975558544031398277480, −7.63801172658218308492118542963, −7.16708748858738772212216473851, −6.08109876301496222359277774549, −5.16156155752868525817998453762, −4.17677202957386955320901384042, −3.39006966306001376982024128932, −1.89850389442155214634578328011, −1.06346747357381321427304379993, 0,
1.06346747357381321427304379993, 1.89850389442155214634578328011, 3.39006966306001376982024128932, 4.17677202957386955320901384042, 5.16156155752868525817998453762, 6.08109876301496222359277774549, 7.16708748858738772212216473851, 7.63801172658218308492118542963, 7.87073897975558544031398277480
