| L(s) = 1 | − 5·5-s − 17·7-s − 30·11-s − 61·13-s + 120·17-s + 43·19-s + 90·23-s + 25·25-s + 90·29-s − 8·31-s + 85·35-s + 317·37-s + 30·41-s + 220·43-s − 180·47-s − 54·49-s + 630·53-s + 150·55-s − 840·59-s + 599·61-s + 305·65-s − 107·67-s − 210·71-s − 421·73-s + 510·77-s − 353·79-s − 1.35e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.917·7-s − 0.822·11-s − 1.30·13-s + 1.71·17-s + 0.519·19-s + 0.815·23-s + 1/5·25-s + 0.576·29-s − 0.0463·31-s + 0.410·35-s + 1.40·37-s + 0.114·41-s + 0.780·43-s − 0.558·47-s − 0.157·49-s + 1.63·53-s + 0.367·55-s − 1.85·59-s + 1.25·61-s + 0.582·65-s − 0.195·67-s − 0.351·71-s − 0.674·73-s + 0.754·77-s − 0.502·79-s − 1.78·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 17 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 61 T + p^{3} T^{2} \) |
| 17 | \( 1 - 120 T + p^{3} T^{2} \) |
| 19 | \( 1 - 43 T + p^{3} T^{2} \) |
| 23 | \( 1 - 90 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 T + p^{3} T^{2} \) |
| 37 | \( 1 - 317 T + p^{3} T^{2} \) |
| 41 | \( 1 - 30 T + p^{3} T^{2} \) |
| 43 | \( 1 - 220 T + p^{3} T^{2} \) |
| 47 | \( 1 + 180 T + p^{3} T^{2} \) |
| 53 | \( 1 - 630 T + p^{3} T^{2} \) |
| 59 | \( 1 + 840 T + p^{3} T^{2} \) |
| 61 | \( 1 - 599 T + p^{3} T^{2} \) |
| 67 | \( 1 + 107 T + p^{3} T^{2} \) |
| 71 | \( 1 + 210 T + p^{3} T^{2} \) |
| 73 | \( 1 + 421 T + p^{3} T^{2} \) |
| 79 | \( 1 + 353 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1350 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 + 997 T + p^{3} T^{2} \) |
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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
−8.167916151846390093977672788655, −7.52471448735218869015941833461, −6.98196299776291780685326666539, −5.85530664796008564118980561115, −5.20172321284862770687339071879, −4.26873995793062394834456541178, −3.10641603143488686593690707782, −2.69944314683935520552314588249, −1.04411892749518207338106677891, 0,
1.04411892749518207338106677891, 2.69944314683935520552314588249, 3.10641603143488686593690707782, 4.26873995793062394834456541178, 5.20172321284862770687339071879, 5.85530664796008564118980561115, 6.98196299776291780685326666539, 7.52471448735218869015941833461, 8.167916151846390093977672788655
