| L(s) = 1 | − 2.29·3-s − 1.78·5-s + 0.671·7-s + 2.25·9-s + 2.23·11-s + 0.0525·13-s + 4.08·15-s + 5.52·17-s + 1.49·19-s − 1.53·21-s − 7.13·23-s − 1.81·25-s + 1.71·27-s + 2.08·29-s − 9.61·31-s − 5.11·33-s − 1.19·35-s + 5.33·37-s − 0.120·39-s − 11.8·41-s − 9.09·43-s − 4.01·45-s + 5.21·47-s − 6.54·49-s − 12.6·51-s + 4.27·53-s − 3.98·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 0.798·5-s + 0.253·7-s + 0.750·9-s + 0.672·11-s + 0.0145·13-s + 1.05·15-s + 1.34·17-s + 0.343·19-s − 0.335·21-s − 1.48·23-s − 0.362·25-s + 0.330·27-s + 0.387·29-s − 1.72·31-s − 0.889·33-s − 0.202·35-s + 0.876·37-s − 0.0192·39-s − 1.85·41-s − 1.38·43-s − 0.598·45-s + 0.760·47-s − 0.935·49-s − 1.77·51-s + 0.587·53-s − 0.536·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 - 0.671T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 0.0525T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 19 | \( 1 - 1.49T + 19T^{2} \) |
| 23 | \( 1 + 7.13T + 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 + 9.61T + 31T^{2} \) |
| 37 | \( 1 - 5.33T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + 0.163T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 8.24T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 7.86T + 79T^{2} \) |
| 83 | \( 1 + 5.22T + 83T^{2} \) |
| 89 | \( 1 + 2.70T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.853553919596045368128290514411, −8.614417304392901343590810120365, −7.76987306491436870778229261547, −6.97410516885834799994845601365, −5.97522032988107134110855398286, −5.35086619202859073536721192701, −4.29417034588159966760518849808, −3.41392202878164219435148698372, −1.48858364007952446800782376760, 0,
1.48858364007952446800782376760, 3.41392202878164219435148698372, 4.29417034588159966760518849808, 5.35086619202859073536721192701, 5.97522032988107134110855398286, 6.97410516885834799994845601365, 7.76987306491436870778229261547, 8.614417304392901343590810120365, 9.853553919596045368128290514411
