| L(s) = 1 | − 5·5-s − 8·7-s + 18·11-s + 8·13-s − 15·17-s − 23·19-s + 63·23-s + 25·25-s − 156·29-s + 85·31-s + 40·35-s + 74·37-s − 246·41-s + 190·43-s + 288·47-s − 279·49-s + 177·53-s − 90·55-s + 792·59-s − 907·61-s − 40·65-s + 322·67-s − 270·71-s + 254·73-s − 144·77-s + 1.12e3·79-s − 771·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.431·7-s + 0.493·11-s + 0.170·13-s − 0.214·17-s − 0.277·19-s + 0.571·23-s + 1/5·25-s − 0.998·29-s + 0.492·31-s + 0.193·35-s + 0.328·37-s − 0.937·41-s + 0.673·43-s + 0.893·47-s − 0.813·49-s + 0.458·53-s − 0.220·55-s + 1.74·59-s − 1.90·61-s − 0.0763·65-s + 0.587·67-s − 0.451·71-s + 0.407·73-s − 0.213·77-s + 1.59·79-s − 1.01·83-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 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 8 T + p^{3} T^{2} \) |
| 17 | \( 1 + 15 T + p^{3} T^{2} \) |
| 19 | \( 1 + 23 T + p^{3} T^{2} \) |
| 23 | \( 1 - 63 T + p^{3} T^{2} \) |
| 29 | \( 1 + 156 T + p^{3} T^{2} \) |
| 31 | \( 1 - 85 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 190 T + p^{3} T^{2} \) |
| 47 | \( 1 - 288 T + p^{3} T^{2} \) |
| 53 | \( 1 - 177 T + p^{3} T^{2} \) |
| 59 | \( 1 - 792 T + p^{3} T^{2} \) |
| 61 | \( 1 + 907 T + p^{3} T^{2} \) |
| 67 | \( 1 - 322 T + p^{3} T^{2} \) |
| 71 | \( 1 + 270 T + p^{3} T^{2} \) |
| 73 | \( 1 - 254 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1123 T + p^{3} T^{2} \) |
| 83 | \( 1 + 771 T + p^{3} T^{2} \) |
| 89 | \( 1 - 198 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1192 T + p^{3} T^{2} \) |
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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
−8.394530659720391593910881659716, −7.51273883078497132589847173444, −6.79201211004997515760888809256, −6.06977919209184883295176594166, −5.10196003005159543742561444950, −4.14820484956074570619756692372, −3.44577284970811415679541799962, −2.40220856989145691980502650955, −1.15838644940166731921020704392, 0,
1.15838644940166731921020704392, 2.40220856989145691980502650955, 3.44577284970811415679541799962, 4.14820484956074570619756692372, 5.10196003005159543742561444950, 6.06977919209184883295176594166, 6.79201211004997515760888809256, 7.51273883078497132589847173444, 8.394530659720391593910881659716
