| L(s) = 1 | − 5·5-s − 14·7-s + 3·11-s + 47·13-s + 39·17-s − 32·19-s − 99·23-s + 25·25-s − 51·29-s − 83·31-s + 70·35-s + 314·37-s + 108·41-s − 299·43-s + 531·47-s − 147·49-s − 564·53-s − 15·55-s + 12·59-s + 230·61-s − 235·65-s + 268·67-s + 120·71-s + 1.10e3·73-s − 42·77-s + 739·79-s + 1.08e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.0822·11-s + 1.00·13-s + 0.556·17-s − 0.386·19-s − 0.897·23-s + 1/5·25-s − 0.326·29-s − 0.480·31-s + 0.338·35-s + 1.39·37-s + 0.411·41-s − 1.06·43-s + 1.64·47-s − 3/7·49-s − 1.46·53-s − 0.0367·55-s + 0.0264·59-s + 0.482·61-s − 0.448·65-s + 0.488·67-s + 0.200·71-s + 1.77·73-s − 0.0621·77-s + 1.05·79-s + 1.43·83-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 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 3 T + p^{3} T^{2} \) |
| 13 | \( 1 - 47 T + p^{3} T^{2} \) |
| 17 | \( 1 - 39 T + p^{3} T^{2} \) |
| 19 | \( 1 + 32 T + p^{3} T^{2} \) |
| 23 | \( 1 + 99 T + p^{3} T^{2} \) |
| 29 | \( 1 + 51 T + p^{3} T^{2} \) |
| 31 | \( 1 + 83 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 108 T + p^{3} T^{2} \) |
| 43 | \( 1 + 299 T + p^{3} T^{2} \) |
| 47 | \( 1 - 531 T + p^{3} T^{2} \) |
| 53 | \( 1 + 564 T + p^{3} T^{2} \) |
| 59 | \( 1 - 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 230 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1106 T + p^{3} T^{2} \) |
| 79 | \( 1 - 739 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1086 T + p^{3} T^{2} \) |
| 89 | \( 1 - 120 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1642 T + p^{3} T^{2} \) |
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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
−8.206039674499400801554573790353, −7.72142226606806063044670802921, −6.61529085269633912618643299202, −6.13453375264893980508031804893, −5.18104058540511342490114445103, −4.00189895270115980673101165090, −3.53443410936500912091243470426, −2.41384144506462781980835384211, −1.13587995431193979124584298365, 0,
1.13587995431193979124584298365, 2.41384144506462781980835384211, 3.53443410936500912091243470426, 4.00189895270115980673101165090, 5.18104058540511342490114445103, 6.13453375264893980508031804893, 6.61529085269633912618643299202, 7.72142226606806063044670802921, 8.206039674499400801554573790353
