| L(s) = 1 | + 2·5-s + 11-s − 5·13-s + 6·17-s − 4·23-s − 25-s + 9·29-s + 4·31-s − 4·37-s − 12·41-s − 4·43-s + 10·47-s − 2·53-s + 2·55-s + 7·59-s − 5·61-s − 10·65-s − 3·67-s + 6·71-s − 4·73-s − 15·79-s − 2·83-s + 12·85-s + 14·89-s + 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.301·11-s − 1.38·13-s + 1.45·17-s − 0.834·23-s − 1/5·25-s + 1.67·29-s + 0.718·31-s − 0.657·37-s − 1.87·41-s − 0.609·43-s + 1.45·47-s − 0.274·53-s + 0.269·55-s + 0.911·59-s − 0.640·61-s − 1.24·65-s − 0.366·67-s + 0.712·71-s − 0.468·73-s − 1.68·79-s − 0.219·83-s + 1.30·85-s + 1.48·89-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.388436608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.388436608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.53717430070209, −12.16028329160302, −11.81650768216451, −11.56202473418954, −10.47302748791253, −10.27715850373846, −9.975559600005008, −9.677265583820266, −8.958041480998367, −8.581953812012494, −7.991586864195594, −7.561358857287053, −7.073695070763455, −6.448723887088507, −6.181733762056821, −5.490080237721162, −5.121797478578575, −4.729769951216627, −3.983283012240840, −3.481858728327131, −2.728914662553280, −2.482839313249199, −1.667376504074404, −1.276798023809428, −0.4012458427306333,
0.4012458427306333, 1.276798023809428, 1.667376504074404, 2.482839313249199, 2.728914662553280, 3.481858728327131, 3.983283012240840, 4.729769951216627, 5.121797478578575, 5.490080237721162, 6.181733762056821, 6.448723887088507, 7.073695070763455, 7.561358857287053, 7.991586864195594, 8.581953812012494, 8.958041480998367, 9.677265583820266, 9.975559600005008, 10.27715850373846, 10.47302748791253, 11.56202473418954, 11.81650768216451, 12.16028329160302, 12.53717430070209
