| L(s) = 1 | − 3-s + 3.45·5-s − 3.47·7-s + 9-s + 3.56·11-s − 1.61·13-s − 3.45·15-s − 3.56·17-s − 5.31·19-s + 3.47·21-s + 6.30·23-s + 6.92·25-s − 27-s − 4.36·29-s − 6.92·31-s − 3.56·33-s − 11.9·35-s + 10.7·37-s + 1.61·39-s + 7.92·41-s − 0.775·43-s + 3.45·45-s − 11.9·47-s + 5.05·49-s + 3.56·51-s − 5.92·53-s + 12.2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.54·5-s − 1.31·7-s + 0.333·9-s + 1.07·11-s − 0.446·13-s − 0.891·15-s − 0.863·17-s − 1.21·19-s + 0.757·21-s + 1.31·23-s + 1.38·25-s − 0.192·27-s − 0.810·29-s − 1.24·31-s − 0.619·33-s − 2.02·35-s + 1.76·37-s + 0.257·39-s + 1.23·41-s − 0.118·43-s + 0.514·45-s − 1.74·47-s + 0.721·49-s + 0.498·51-s − 0.813·53-s + 1.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 5 | \( 1 - 3.45T + 5T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 + 5.31T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 7.92T + 41T^{2} \) |
| 43 | \( 1 + 0.775T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 5.92T + 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 - 5.03T + 61T^{2} \) |
| 67 | \( 1 + 0.281T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 3.01T + 73T^{2} \) |
| 79 | \( 1 + 0.622T + 79T^{2} \) |
| 83 | \( 1 + 8.17T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−7.34143856957826996981727778384, −6.75112153020968495831256010111, −6.16159739718544899802822946518, −5.90392961941376554821699197661, −4.87098833525889664087728743826, −4.13747507594807290006326258005, −3.08557312843631787727351227910, −2.24314883581745059315648645994, −1.36765696615734593232143783815, 0,
1.36765696615734593232143783815, 2.24314883581745059315648645994, 3.08557312843631787727351227910, 4.13747507594807290006326258005, 4.87098833525889664087728743826, 5.90392961941376554821699197661, 6.16159739718544899802822946518, 6.75112153020968495831256010111, 7.34143856957826996981727778384
