| L(s) = 1 | − 5-s + 4.34·7-s + 2.70·11-s − 6.26·13-s + 5.37·17-s + 6.23·19-s + 5.61·23-s + 25-s − 0.762·29-s − 6.61·31-s − 4.34·35-s + 11.1·37-s − 9.02·41-s − 9.68·43-s + 5.22·47-s + 11.9·49-s + 10.6·53-s − 2.70·55-s + 1.73·59-s + 1.63·61-s + 6.26·65-s − 9.52·67-s + 14.3·71-s − 10.6·73-s + 11.7·77-s − 0.535·79-s + 0.146·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.64·7-s + 0.814·11-s − 1.73·13-s + 1.30·17-s + 1.43·19-s + 1.17·23-s + 0.200·25-s − 0.141·29-s − 1.18·31-s − 0.735·35-s + 1.83·37-s − 1.40·41-s − 1.47·43-s + 0.762·47-s + 1.70·49-s + 1.46·53-s − 0.364·55-s + 0.225·59-s + 0.209·61-s + 0.777·65-s − 1.16·67-s + 1.69·71-s − 1.24·73-s + 1.33·77-s − 0.0602·79-s + 0.0160·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.529641632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.529641632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 + 6.26T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 - 5.61T + 23T^{2} \) |
| 29 | \( 1 + 0.762T + 29T^{2} \) |
| 31 | \( 1 + 6.61T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 + 9.68T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 - 1.63T + 61T^{2} \) |
| 67 | \( 1 + 9.52T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 0.535T + 79T^{2} \) |
| 83 | \( 1 - 0.146T + 83T^{2} \) |
| 89 | \( 1 - 7.85T + 89T^{2} \) |
| 97 | \( 1 + 0.389T + 97T^{2} \) |
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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
−7.85499251293508735868834875172, −7.39539809527383533148560127157, −6.94858545636073399421985010858, −5.55917499103403814837117706331, −5.16408713449294106429679578114, −4.55073464665465719991411251167, −3.63682327774320897440643295655, −2.77076629632220862226554664254, −1.69225860155923813715176337943, −0.887501263415251068237805751718,
0.887501263415251068237805751718, 1.69225860155923813715176337943, 2.77076629632220862226554664254, 3.63682327774320897440643295655, 4.55073464665465719991411251167, 5.16408713449294106429679578114, 5.55917499103403814837117706331, 6.94858545636073399421985010858, 7.39539809527383533148560127157, 7.85499251293508735868834875172
