| L(s) = 1 | − 3.29·3-s + 3.07·5-s − 7-s + 7.87·9-s − 5.34·11-s − 5.07·13-s − 10.1·15-s − 3.85·17-s + 3.78·19-s + 3.29·21-s + 3.51·23-s + 4.42·25-s − 16.0·27-s + 6.93·29-s + 4.21·31-s + 17.6·33-s − 3.07·35-s − 0.205·37-s + 16.7·39-s + 8.93·41-s − 3.22·43-s + 24.1·45-s − 0.0916·47-s + 49-s + 12.6·51-s − 3.19·53-s − 16.4·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 1.37·5-s − 0.377·7-s + 2.62·9-s − 1.61·11-s − 1.40·13-s − 2.61·15-s − 0.933·17-s + 0.869·19-s + 0.719·21-s + 0.732·23-s + 0.885·25-s − 3.09·27-s + 1.28·29-s + 0.756·31-s + 3.06·33-s − 0.518·35-s − 0.0337·37-s + 2.68·39-s + 1.39·41-s − 0.492·43-s + 3.60·45-s − 0.0133·47-s + 0.142·49-s + 1.77·51-s − 0.438·53-s − 2.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 3 | \( 1 + 3.29T + 3T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 + 0.205T + 37T^{2} \) |
| 41 | \( 1 - 8.93T + 41T^{2} \) |
| 43 | \( 1 + 3.22T + 43T^{2} \) |
| 47 | \( 1 + 0.0916T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 + 2.87T + 61T^{2} \) |
| 67 | \( 1 - 2.16T + 67T^{2} \) |
| 71 | \( 1 - 0.761T + 71T^{2} \) |
| 73 | \( 1 + 0.558T + 73T^{2} \) |
| 79 | \( 1 + 6.27T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 0.298T + 89T^{2} \) |
| 97 | \( 1 - 9.62T + 97T^{2} \) |
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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
−7.12739146409989389622469019821, −6.86101786349739715342146880655, −5.93437121474922933938034925236, −5.59026069014485200071413539771, −4.86365898083646148057575942822, −4.57611277208408617849747956642, −2.86756742676481588833164248897, −2.22064950921836214881050028727, −1.02477776815809675764341897778, 0,
1.02477776815809675764341897778, 2.22064950921836214881050028727, 2.86756742676481588833164248897, 4.57611277208408617849747956642, 4.86365898083646148057575942822, 5.59026069014485200071413539771, 5.93437121474922933938034925236, 6.86101786349739715342146880655, 7.12739146409989389622469019821
