| L(s) = 1 | − 2-s + 4-s + 3.18·5-s − 3.18·7-s − 8-s − 3.18·10-s − 3.18·11-s − 2.56·13-s + 3.18·14-s + 16-s − 2·17-s + 4.74·19-s + 3.18·20-s + 3.18·22-s + 6.93·23-s + 5.13·25-s + 2.56·26-s − 3.18·28-s − 10.1·29-s + 2.43·31-s − 32-s + 2·34-s − 10.1·35-s − 4.74·37-s − 4.74·38-s − 3.18·40-s − 41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.42·5-s − 1.20·7-s − 0.353·8-s − 1.00·10-s − 0.959·11-s − 0.711·13-s + 0.850·14-s + 0.250·16-s − 0.485·17-s + 1.08·19-s + 0.711·20-s + 0.678·22-s + 1.44·23-s + 1.02·25-s + 0.503·26-s − 0.601·28-s − 1.89·29-s + 0.437·31-s − 0.176·32-s + 0.342·34-s − 1.71·35-s − 0.780·37-s − 0.770·38-s − 0.503·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 5.61T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 9.87T + 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 - 8.24T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 8.98T + 97T^{2} \) |
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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
−9.043947351209761195094562866071, −7.84111288222427836742114043914, −7.09815502580848816133797920626, −6.41050265507106109942546562679, −5.60372409793456316728915840105, −4.96268227894045167064751909168, −3.25573466520558379644566371901, −2.64356150539194671726985106488, −1.59514033818002323082254662873, 0,
1.59514033818002323082254662873, 2.64356150539194671726985106488, 3.25573466520558379644566371901, 4.96268227894045167064751909168, 5.60372409793456316728915840105, 6.41050265507106109942546562679, 7.09815502580848816133797920626, 7.84111288222427836742114043914, 9.043947351209761195094562866071
