| L(s) = 1 | + 2.50·3-s + 5-s − 2.23·7-s + 3.25·9-s − 0.595·11-s − 3.72·13-s + 2.50·15-s − 2.87·17-s − 4.54·19-s − 5.58·21-s + 6.52·23-s + 25-s + 0.634·27-s − 3.40·29-s − 4.46·31-s − 1.48·33-s − 2.23·35-s − 1.19·37-s − 9.31·39-s + 6.99·41-s − 6.50·43-s + 3.25·45-s + 1.73·47-s − 2.00·49-s − 7.19·51-s + 1.40·53-s − 0.595·55-s + ⋯ |
| L(s) = 1 | + 1.44·3-s + 0.447·5-s − 0.844·7-s + 1.08·9-s − 0.179·11-s − 1.03·13-s + 0.645·15-s − 0.697·17-s − 1.04·19-s − 1.21·21-s + 1.35·23-s + 0.200·25-s + 0.122·27-s − 0.632·29-s − 0.801·31-s − 0.259·33-s − 0.377·35-s − 0.197·37-s − 1.49·39-s + 1.09·41-s − 0.991·43-s + 0.485·45-s + 0.252·47-s − 0.286·49-s − 1.00·51-s + 0.192·53-s − 0.0803·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 + 0.595T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 + 4.54T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 3.40T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 1.19T + 37T^{2} \) |
| 41 | \( 1 - 6.99T + 41T^{2} \) |
| 43 | \( 1 + 6.50T + 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 - 4.59T + 59T^{2} \) |
| 61 | \( 1 + 0.884T + 61T^{2} \) |
| 67 | \( 1 + 7.08T + 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 5.17T + 79T^{2} \) |
| 83 | \( 1 + 1.25T + 83T^{2} \) |
| 89 | \( 1 + 0.496T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−7.67599810439103221898042932743, −7.15714776073075401770738406504, −6.49404386003611836324481817773, −5.58422677153683748734210399718, −4.67365912486107403837117886806, −3.87990357094745073975048006281, −2.99331352791671227635514808124, −2.52081227440498712392612148022, −1.70102033360078356689653026280, 0,
1.70102033360078356689653026280, 2.52081227440498712392612148022, 2.99331352791671227635514808124, 3.87990357094745073975048006281, 4.67365912486107403837117886806, 5.58422677153683748734210399718, 6.49404386003611836324481817773, 7.15714776073075401770738406504, 7.67599810439103221898042932743
