| L(s) = 1 | + 3.37·3-s + 5-s − 3.80·7-s + 8.37·9-s − 2.57·11-s − 4.49·13-s + 3.37·15-s + 4.43·17-s − 6.71·19-s − 12.8·21-s − 8.16·23-s + 25-s + 18.1·27-s − 8.65·29-s − 2.63·31-s − 8.69·33-s − 3.80·35-s − 2.00·37-s − 15.1·39-s + 0.367·41-s − 4.23·43-s + 8.37·45-s − 6.96·47-s + 7.48·49-s + 14.9·51-s + 12.5·53-s − 2.57·55-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 0.447·5-s − 1.43·7-s + 2.79·9-s − 0.776·11-s − 1.24·13-s + 0.870·15-s + 1.07·17-s − 1.54·19-s − 2.80·21-s − 1.70·23-s + 0.200·25-s + 3.49·27-s − 1.60·29-s − 0.473·31-s − 1.51·33-s − 0.643·35-s − 0.329·37-s − 2.42·39-s + 0.0573·41-s − 0.645·43-s + 1.24·45-s − 1.01·47-s + 1.06·49-s + 2.09·51-s + 1.72·53-s − 0.347·55-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 | \( 1 - 3.37T + 3T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 4.43T + 17T^{2} \) |
| 19 | \( 1 + 6.71T + 19T^{2} \) |
| 23 | \( 1 + 8.16T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 + 2.63T + 31T^{2} \) |
| 37 | \( 1 + 2.00T + 37T^{2} \) |
| 41 | \( 1 - 0.367T + 41T^{2} \) |
| 43 | \( 1 + 4.23T + 43T^{2} \) |
| 47 | \( 1 + 6.96T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 2.01T + 59T^{2} \) |
| 61 | \( 1 + 6.13T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 - 0.138T + 73T^{2} \) |
| 79 | \( 1 - 6.47T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 7.03T + 97T^{2} \) |
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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
−7.72701819504685875985344803479, −7.29047167571958246352272299530, −6.47568022283739267404125478169, −5.64959225465095524149803297128, −4.56803943322035457059269495770, −3.70800359582127848818096898664, −3.19780283020382314112487015400, −2.33041584617927490029306723576, −1.91314137230346783544968366268, 0,
1.91314137230346783544968366268, 2.33041584617927490029306723576, 3.19780283020382314112487015400, 3.70800359582127848818096898664, 4.56803943322035457059269495770, 5.64959225465095524149803297128, 6.47568022283739267404125478169, 7.29047167571958246352272299530, 7.72701819504685875985344803479
