| L(s) = 1 | + 2.35·3-s − 3.32·5-s + 1.00·7-s + 2.56·9-s − 0.281·11-s + 0.731·13-s − 7.83·15-s + 4.74·17-s − 5.22·19-s + 2.36·21-s − 6.35·23-s + 6.03·25-s − 1.02·27-s + 2.12·29-s + 1.17·31-s − 0.664·33-s − 3.32·35-s + 1.39·37-s + 1.72·39-s − 2.17·41-s − 11.2·43-s − 8.52·45-s + 2.59·47-s − 5.99·49-s + 11.2·51-s + 9.36·53-s + 0.935·55-s + ⋯ |
| L(s) = 1 | + 1.36·3-s − 1.48·5-s + 0.378·7-s + 0.855·9-s − 0.0848·11-s + 0.202·13-s − 2.02·15-s + 1.15·17-s − 1.19·19-s + 0.515·21-s − 1.32·23-s + 1.20·25-s − 0.196·27-s + 0.394·29-s + 0.210·31-s − 0.115·33-s − 0.562·35-s + 0.230·37-s + 0.276·39-s − 0.340·41-s − 1.71·43-s − 1.27·45-s + 0.377·47-s − 0.856·49-s + 1.56·51-s + 1.28·53-s + 0.126·55-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 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 + 0.281T + 11T^{2} \) |
| 13 | \( 1 - 0.731T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 + 5.22T + 19T^{2} \) |
| 23 | \( 1 + 6.35T + 23T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 2.59T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 2.15T + 67T^{2} \) |
| 71 | \( 1 - 4.52T + 71T^{2} \) |
| 73 | \( 1 - 3.16T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 2.06T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.976071172573929816494504711376, −7.34716122128180009188429117368, −6.53176507087284386785955633894, −5.49305226623947589534784628129, −4.45388498076034497421458588842, −3.91269385997177037814726473857, −3.33196042194176676842389509876, −2.50108156780215802945501460728, −1.48389686407987205519219669228, 0,
1.48389686407987205519219669228, 2.50108156780215802945501460728, 3.33196042194176676842389509876, 3.91269385997177037814726473857, 4.45388498076034497421458588842, 5.49305226623947589534784628129, 6.53176507087284386785955633894, 7.34716122128180009188429117368, 7.976071172573929816494504711376
