| L(s) = 1 | − 0.745·3-s + 0.935·5-s + 7-s − 2.44·9-s − 0.508·11-s − 5.36·13-s − 0.697·15-s − 17-s − 5.87·19-s − 0.745·21-s + 3.36·23-s − 4.12·25-s + 4.06·27-s + 1.87·29-s + 3.91·31-s + 0.379·33-s + 0.935·35-s − 2.12·37-s + 4·39-s − 9.12·41-s − 2.10·43-s − 2.28·45-s − 2.98·47-s + 49-s + 0.745·51-s − 11.3·53-s − 0.475·55-s + ⋯ |
| L(s) = 1 | − 0.430·3-s + 0.418·5-s + 0.377·7-s − 0.814·9-s − 0.153·11-s − 1.48·13-s − 0.180·15-s − 0.242·17-s − 1.34·19-s − 0.162·21-s + 0.701·23-s − 0.824·25-s + 0.781·27-s + 0.347·29-s + 0.703·31-s + 0.0659·33-s + 0.158·35-s − 0.350·37-s + 0.640·39-s − 1.42·41-s − 0.321·43-s − 0.340·45-s − 0.435·47-s + 0.142·49-s + 0.104·51-s − 1.55·53-s − 0.0641·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.745T + 3T^{2} \) |
| 5 | \( 1 - 0.935T + 5T^{2} \) |
| 11 | \( 1 + 0.508T + 11T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 19 | \( 1 + 5.87T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 - 3.91T + 31T^{2} \) |
| 37 | \( 1 + 2.12T + 37T^{2} \) |
| 41 | \( 1 + 9.12T + 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 + 9.25T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 + 3.83T + 71T^{2} \) |
| 73 | \( 1 - 6.14T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 9.87T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 9.28T + 97T^{2} \) |
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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
−9.741880458752732798168912215500, −8.741982756997234102511269938551, −8.034967079381806913777910369105, −6.94310935114716150109795689340, −6.15904428139229912113815858097, −5.18491043485289062408372557616, −4.54742915557564384178882243886, −2.98226586704130845961617796600, −1.96564889349857540777405033955, 0,
1.96564889349857540777405033955, 2.98226586704130845961617796600, 4.54742915557564384178882243886, 5.18491043485289062408372557616, 6.15904428139229912113815858097, 6.94310935114716150109795689340, 8.034967079381806913777910369105, 8.741982756997234102511269938551, 9.741880458752732798168912215500
