| L(s) = 1 | + 2.52·3-s + 2.44·5-s − 4.18·7-s + 3.35·9-s − 1.67·11-s − 2.61·13-s + 6.16·15-s − 4.89·17-s + 5.68·19-s − 10.5·21-s − 9.11·23-s + 0.980·25-s + 0.899·27-s − 0.674·29-s + 6.86·31-s − 4.22·33-s − 10.2·35-s − 9.66·37-s − 6.58·39-s + 11.1·41-s − 2.46·43-s + 8.20·45-s + 1.09·47-s + 10.4·49-s − 12.3·51-s + 0.485·53-s − 4.09·55-s + ⋯ |
| L(s) = 1 | + 1.45·3-s + 1.09·5-s − 1.58·7-s + 1.11·9-s − 0.505·11-s − 0.724·13-s + 1.59·15-s − 1.18·17-s + 1.30·19-s − 2.30·21-s − 1.90·23-s + 0.196·25-s + 0.173·27-s − 0.125·29-s + 1.23·31-s − 0.735·33-s − 1.72·35-s − 1.58·37-s − 1.05·39-s + 1.73·41-s − 0.376·43-s + 1.22·45-s + 0.159·47-s + 1.49·49-s − 1.72·51-s + 0.0667·53-s − 0.552·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 2.52T + 3T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 5.68T + 19T^{2} \) |
| 23 | \( 1 + 9.11T + 23T^{2} \) |
| 29 | \( 1 + 0.674T + 29T^{2} \) |
| 31 | \( 1 - 6.86T + 31T^{2} \) |
| 37 | \( 1 + 9.66T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 2.46T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 - 0.485T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 9.99T + 67T^{2} \) |
| 71 | \( 1 + 6.25T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 4.67T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 5.29T + 89T^{2} \) |
| 97 | \( 1 + 4.82T + 97T^{2} \) |
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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
−7.67813938001668676315631297976, −7.19287630798032270101663015803, −6.21677714530501655217241416793, −5.85069167755864024453572938469, −4.70648093174141676761854905512, −3.80015933498018093999294568494, −2.96128886973588037677250279691, −2.53104706233966701232450887211, −1.75298539733877271806614227546, 0,
1.75298539733877271806614227546, 2.53104706233966701232450887211, 2.96128886973588037677250279691, 3.80015933498018093999294568494, 4.70648093174141676761854905512, 5.85069167755864024453572938469, 6.21677714530501655217241416793, 7.19287630798032270101663015803, 7.67813938001668676315631297976
