| L(s) = 1 | + 19.8·5-s + 5.87·7-s + 18.7·11-s + 45.8·13-s − 16.8·17-s + 10.3·19-s + 49.8·23-s + 269.·25-s − 10.9·29-s − 151.·31-s + 116.·35-s + 346.·37-s − 264.·41-s + 411.·43-s + 472.·47-s − 308.·49-s − 290.·53-s + 372.·55-s + 53.2·59-s − 293.·61-s + 911.·65-s + 398.·67-s − 647.·71-s − 478.·73-s + 110.·77-s − 374.·79-s + 933.·83-s + ⋯ |
| L(s) = 1 | + 1.77·5-s + 0.316·7-s + 0.513·11-s + 0.978·13-s − 0.240·17-s + 0.124·19-s + 0.452·23-s + 2.15·25-s − 0.0698·29-s − 0.878·31-s + 0.563·35-s + 1.53·37-s − 1.00·41-s + 1.46·43-s + 1.46·47-s − 0.899·49-s − 0.752·53-s + 0.912·55-s + 0.117·59-s − 0.616·61-s + 1.73·65-s + 0.726·67-s − 1.08·71-s − 0.766·73-s + 0.162·77-s − 0.533·79-s + 1.23·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.750518920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.750518920\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 19.8T + 125T^{2} \) |
| 7 | \( 1 - 5.87T + 343T^{2} \) |
| 11 | \( 1 - 18.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 264.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 411.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 472.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 53.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 293.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 398.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 647.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 478.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 374.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 933.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 368.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 274.T + 9.12e5T^{2} \) |
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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
−9.196161750803852163774445636738, −8.819870043880222730226630586316, −7.62036449804128574326020507338, −6.56155029722087703891060265418, −5.98975605576082532514490294539, −5.26220346386527085954813678310, −4.18527119308553829259943158599, −2.91398181979157011150304875709, −1.87441850799523545603158280140, −1.07367956835293208359926476904,
1.07367956835293208359926476904, 1.87441850799523545603158280140, 2.91398181979157011150304875709, 4.18527119308553829259943158599, 5.26220346386527085954813678310, 5.98975605576082532514490294539, 6.56155029722087703891060265418, 7.62036449804128574326020507338, 8.819870043880222730226630586316, 9.196161750803852163774445636738
