| L(s) = 1 | − 2-s + 4-s − 0.580·5-s − 1.39·7-s − 8-s + 0.580·10-s + 3.94·11-s − 5.15·13-s + 1.39·14-s + 16-s + 1.15·17-s − 6.17·19-s − 0.580·20-s − 3.94·22-s − 8.81·23-s − 4.66·25-s + 5.15·26-s − 1.39·28-s + 3.09·29-s + 4.25·31-s − 32-s − 1.15·34-s + 0.810·35-s + 11.3·37-s + 6.17·38-s + 0.580·40-s + 2.30·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.259·5-s − 0.527·7-s − 0.353·8-s + 0.183·10-s + 1.19·11-s − 1.43·13-s + 0.372·14-s + 0.250·16-s + 0.279·17-s − 1.41·19-s − 0.129·20-s − 0.841·22-s − 1.83·23-s − 0.932·25-s + 1.01·26-s − 0.263·28-s + 0.575·29-s + 0.765·31-s − 0.176·32-s − 0.197·34-s + 0.136·35-s + 1.87·37-s + 1.00·38-s + 0.0918·40-s + 0.359·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8641041152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8641041152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 0.580T + 5T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 23 | \( 1 + 8.81T + 23T^{2} \) |
| 29 | \( 1 - 3.09T + 29T^{2} \) |
| 31 | \( 1 - 4.25T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 - 6.45T + 43T^{2} \) |
| 47 | \( 1 - 4.58T + 47T^{2} \) |
| 53 | \( 1 + 7.69T + 53T^{2} \) |
| 59 | \( 1 - 8.70T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 8.42T + 71T^{2} \) |
| 73 | \( 1 + 4.40T + 73T^{2} \) |
| 79 | \( 1 - 2.59T + 79T^{2} \) |
| 83 | \( 1 + 6.07T + 83T^{2} \) |
| 89 | \( 1 + 2.89T + 89T^{2} \) |
| 97 | \( 1 + 3.84T + 97T^{2} \) |
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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
−8.332426920344239027714627256524, −7.87959159984485116465700802329, −7.01937013159502480366800993319, −6.35125729165426493697783839352, −5.79279696531559785255047214645, −4.40378254410386503392233487986, −3.97067641955837568557949526550, −2.70402052516242545643271379698, −1.97458335718549929429175228531, −0.57093366483578033172605006334,
0.57093366483578033172605006334, 1.97458335718549929429175228531, 2.70402052516242545643271379698, 3.97067641955837568557949526550, 4.40378254410386503392233487986, 5.79279696531559785255047214645, 6.35125729165426493697783839352, 7.01937013159502480366800993319, 7.87959159984485116465700802329, 8.332426920344239027714627256524
