| L(s) = 1 | + 2.24·3-s + 4.24·5-s + 7-s + 2.05·9-s + 1.05·11-s + 0.941·13-s + 9.55·15-s + 5.30·17-s − 5.55·19-s + 2.24·21-s + 23-s + 13.0·25-s − 2.11·27-s − 9.43·29-s + 3.30·31-s + 2.38·33-s + 4.24·35-s − 8.61·37-s + 2.11·39-s − 9.11·41-s + 1.88·43-s + 8.74·45-s − 3.30·47-s + 49-s + 11.9·51-s + 4.61·53-s + 4.49·55-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 1.90·5-s + 0.377·7-s + 0.686·9-s + 0.319·11-s + 0.261·13-s + 2.46·15-s + 1.28·17-s − 1.27·19-s + 0.490·21-s + 0.208·23-s + 2.61·25-s − 0.407·27-s − 1.75·29-s + 0.594·31-s + 0.414·33-s + 0.718·35-s − 1.41·37-s + 0.339·39-s − 1.42·41-s + 0.287·43-s + 1.30·45-s − 0.482·47-s + 0.142·49-s + 1.67·51-s + 0.633·53-s + 0.606·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.335557676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.335557676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 0.941T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 + 5.55T + 19T^{2} \) |
| 29 | \( 1 + 9.43T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 - 4.61T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 + 6.13T + 61T^{2} \) |
| 67 | \( 1 + 1.05T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 6.73T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 9.05T + 83T^{2} \) |
| 89 | \( 1 - 3.19T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−8.801460276775307059830146292340, −8.470959579861513984217480417866, −7.43252304720806733917885500284, −6.58177241662382855330673833962, −5.75085255161351736683985147323, −5.12154746514096557906382084904, −3.85730136607479955519816011553, −2.98990696425691616777739601118, −2.04880401891516465083716862170, −1.52512056153396523192106609558,
1.52512056153396523192106609558, 2.04880401891516465083716862170, 2.98990696425691616777739601118, 3.85730136607479955519816011553, 5.12154746514096557906382084904, 5.75085255161351736683985147323, 6.58177241662382855330673833962, 7.43252304720806733917885500284, 8.470959579861513984217480417866, 8.801460276775307059830146292340
