| L(s) = 1 | − 14·5-s + 36·7-s + 36·11-s − 54·13-s + 22·17-s + 36·19-s − 144·23-s + 71·25-s + 50·29-s + 108·31-s − 504·35-s − 214·37-s + 446·41-s + 252·43-s + 72·47-s + 953·49-s − 22·53-s − 504·55-s + 684·59-s + 466·61-s + 756·65-s − 180·67-s + 576·71-s − 54·73-s + 1.29e3·77-s + 972·79-s + 684·83-s + ⋯ |
| L(s) = 1 | − 1.25·5-s + 1.94·7-s + 0.986·11-s − 1.15·13-s + 0.313·17-s + 0.434·19-s − 1.30·23-s + 0.567·25-s + 0.320·29-s + 0.625·31-s − 2.43·35-s − 0.950·37-s + 1.69·41-s + 0.893·43-s + 0.223·47-s + 2.77·49-s − 0.0570·53-s − 1.23·55-s + 1.50·59-s + 0.978·61-s + 1.44·65-s − 0.328·67-s + 0.962·71-s − 0.0865·73-s + 1.91·77-s + 1.38·79-s + 0.904·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.957185317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.957185317\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 14 T + p^{3} T^{2} \) |
| 7 | \( 1 - 36 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 144 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 446 T + p^{3} T^{2} \) |
| 43 | \( 1 - 252 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 22 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 466 T + p^{3} T^{2} \) |
| 67 | \( 1 + 180 T + p^{3} T^{2} \) |
| 71 | \( 1 - 576 T + p^{3} T^{2} \) |
| 73 | \( 1 + 54 T + p^{3} T^{2} \) |
| 79 | \( 1 - 972 T + p^{3} T^{2} \) |
| 83 | \( 1 - 684 T + p^{3} T^{2} \) |
| 89 | \( 1 + 346 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1134 T + p^{3} T^{2} \) |
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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
−10.49665095035779121332201607282, −9.351715057892642451421172170967, −8.240204014667663813313145554331, −7.82393801538496096426939173691, −7.01648431044743300263160069316, −5.50702000463531874435227522166, −4.52401054784339748170566839349, −3.89753289386144511483107132125, −2.21716239707985048478691446911, −0.875461239189174627546007843081,
0.875461239189174627546007843081, 2.21716239707985048478691446911, 3.89753289386144511483107132125, 4.52401054784339748170566839349, 5.50702000463531874435227522166, 7.01648431044743300263160069316, 7.82393801538496096426939173691, 8.240204014667663813313145554331, 9.351715057892642451421172170967, 10.49665095035779121332201607282
