| L(s) = 1 | + 2.69·2-s + 5.28·4-s − 1.58·5-s + 8.87·8-s − 4.28·10-s + 0.300·11-s + 2.81·13-s + 13.3·16-s + 5.87·17-s − 2.28·19-s − 8.39·20-s + 0.810·22-s + 1.88·23-s − 2.47·25-s + 7.58·26-s + 2.52·29-s − 4.81·31-s + 18.3·32-s + 15.8·34-s − 4.47·37-s − 6.17·38-s − 14.0·40-s + 8.90·41-s − 9.09·43-s + 1.58·44-s + 5.09·46-s − 3.21·47-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.64·4-s − 0.710·5-s + 3.13·8-s − 1.35·10-s + 0.0905·11-s + 0.779·13-s + 3.34·16-s + 1.42·17-s − 0.524·19-s − 1.87·20-s + 0.172·22-s + 0.393·23-s − 0.495·25-s + 1.48·26-s + 0.468·29-s − 0.864·31-s + 3.25·32-s + 2.72·34-s − 0.736·37-s − 1.00·38-s − 2.22·40-s + 1.39·41-s − 1.38·43-s + 0.239·44-s + 0.751·46-s − 0.468·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.230223919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.230223919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 - 0.300T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 - 1.88T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 + 4.81T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 - 2.01T + 53T^{2} \) |
| 59 | \( 1 + 4.88T + 59T^{2} \) |
| 61 | \( 1 - 7.57T + 61T^{2} \) |
| 67 | \( 1 - 0.712T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 + 9.35T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−9.970965561289344786049754633346, −8.537460152035363919632972456131, −7.66325265073029221620547312025, −6.96524260346199528915031208320, −6.02816357536300537066653911427, −5.38846002982378239464868949145, −4.37893808747865069366507247351, −3.67695591948767786039385226678, −2.96470202007297714621794415478, −1.55674375407619957778904658424,
1.55674375407619957778904658424, 2.96470202007297714621794415478, 3.67695591948767786039385226678, 4.37893808747865069366507247351, 5.38846002982378239464868949145, 6.02816357536300537066653911427, 6.96524260346199528915031208320, 7.66325265073029221620547312025, 8.537460152035363919632972456131, 9.970965561289344786049754633346
