| L(s) = 1 | + 3.31·3-s + 1.60·5-s + 4.11·7-s + 8.00·9-s + 11-s − 0.192·13-s + 5.33·15-s − 7.62·17-s + 19-s + 13.6·21-s − 6.73·23-s − 2.41·25-s + 16.5·27-s + 6.80·29-s − 7.33·31-s + 3.31·33-s + 6.61·35-s + 0.734·37-s − 0.637·39-s + 0.976·41-s + 5.55·43-s + 12.8·45-s − 8.83·47-s + 9.90·49-s − 25.2·51-s + 11.8·53-s + 1.60·55-s + ⋯ |
| L(s) = 1 | + 1.91·3-s + 0.719·5-s + 1.55·7-s + 2.66·9-s + 0.301·11-s − 0.0533·13-s + 1.37·15-s − 1.84·17-s + 0.229·19-s + 2.97·21-s − 1.40·23-s − 0.482·25-s + 3.19·27-s + 1.26·29-s − 1.31·31-s + 0.577·33-s + 1.11·35-s + 0.120·37-s − 0.102·39-s + 0.152·41-s + 0.847·43-s + 1.91·45-s − 1.28·47-s + 1.41·49-s − 3.54·51-s + 1.62·53-s + 0.216·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.201872712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.201872712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 - 1.60T + 5T^{2} \) |
| 7 | \( 1 - 4.11T + 7T^{2} \) |
| 13 | \( 1 + 0.192T + 13T^{2} \) |
| 17 | \( 1 + 7.62T + 17T^{2} \) |
| 23 | \( 1 + 6.73T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 7.33T + 31T^{2} \) |
| 37 | \( 1 - 0.734T + 37T^{2} \) |
| 41 | \( 1 - 0.976T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 + 8.83T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 3.30T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 4.12T + 71T^{2} \) |
| 73 | \( 1 + 6.15T + 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 + 6.33T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 4.33T + 97T^{2} \) |
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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
−8.592260489303283802131624402473, −8.098534286665835193273947798432, −7.37859430068931387664803977766, −6.62865231023012081633116396482, −5.49463369407200721614688417658, −4.35243355689214477204737166347, −4.12396708757014634792821579990, −2.76082435602366985662187838653, −2.03310294096895119555916590877, −1.56096570250052405401613788032,
1.56096570250052405401613788032, 2.03310294096895119555916590877, 2.76082435602366985662187838653, 4.12396708757014634792821579990, 4.35243355689214477204737166347, 5.49463369407200721614688417658, 6.62865231023012081633116396482, 7.37859430068931387664803977766, 8.098534286665835193273947798432, 8.592260489303283802131624402473
