| L(s) = 1 | − 4.73·3-s − 18.0·5-s − 0.213·7-s − 4.58·9-s + 3.39·11-s − 90.7·13-s + 85.5·15-s − 2.59·17-s − 19·19-s + 1.01·21-s − 26.6·23-s + 201.·25-s + 149.·27-s + 60.1·29-s + 176.·31-s − 16.0·33-s + 3.85·35-s − 154.·37-s + 429.·39-s + 434.·41-s + 365.·43-s + 82.8·45-s − 204.·47-s − 342.·49-s + 12.2·51-s − 135.·53-s − 61.3·55-s + ⋯ |
| L(s) = 1 | − 0.911·3-s − 1.61·5-s − 0.0115·7-s − 0.169·9-s + 0.0930·11-s − 1.93·13-s + 1.47·15-s − 0.0370·17-s − 0.229·19-s + 0.0104·21-s − 0.241·23-s + 1.61·25-s + 1.06·27-s + 0.384·29-s + 1.02·31-s − 0.0847·33-s + 0.0186·35-s − 0.684·37-s + 1.76·39-s + 1.65·41-s + 1.29·43-s + 0.274·45-s − 0.633·47-s − 0.999·49-s + 0.0337·51-s − 0.351·53-s − 0.150·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4356712309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4356712309\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 4.73T + 27T^{2} \) |
| 5 | \( 1 + 18.0T + 125T^{2} \) |
| 7 | \( 1 + 0.213T + 343T^{2} \) |
| 11 | \( 1 - 3.39T + 1.33e3T^{2} \) |
| 13 | \( 1 + 90.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.59T + 4.91e3T^{2} \) |
| 23 | \( 1 + 26.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 60.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 135.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 759.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 284.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 590.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 972.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 368.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 204.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 782.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 213.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−11.38706949386912544181039078755, −10.66335839591820873809749102252, −9.480228984253261288386880547318, −8.182660064137815763193007904729, −7.45115180981913637410561665473, −6.44266419423815731733596920049, −5.06480889553315385568676961501, −4.30108937850090138001115544203, −2.82215607317539289441585266022, −0.45539880345220226819484374287,
0.45539880345220226819484374287, 2.82215607317539289441585266022, 4.30108937850090138001115544203, 5.06480889553315385568676961501, 6.44266419423815731733596920049, 7.45115180981913637410561665473, 8.182660064137815763193007904729, 9.480228984253261288386880547318, 10.66335839591820873809749102252, 11.38706949386912544181039078755
