| L(s) = 1 | − 5·5-s + 12·7-s − 24·11-s + 38·13-s + 6·17-s − 104·19-s + 100·23-s + 25·25-s − 230·29-s + 56·31-s − 60·35-s + 190·37-s − 202·41-s + 148·43-s + 124·47-s − 199·49-s − 206·53-s + 120·55-s − 128·59-s + 190·61-s − 190·65-s + 204·67-s − 440·71-s + 1.21e3·73-s − 288·77-s − 816·79-s − 1.41e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.647·7-s − 0.657·11-s + 0.810·13-s + 0.0856·17-s − 1.25·19-s + 0.906·23-s + 1/5·25-s − 1.47·29-s + 0.324·31-s − 0.289·35-s + 0.844·37-s − 0.769·41-s + 0.524·43-s + 0.384·47-s − 0.580·49-s − 0.533·53-s + 0.294·55-s − 0.282·59-s + 0.398·61-s − 0.362·65-s + 0.371·67-s − 0.735·71-s + 1.93·73-s − 0.426·77-s − 1.16·79-s − 1.86·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 104 T + p^{3} T^{2} \) |
| 23 | \( 1 - 100 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 - 56 T + p^{3} T^{2} \) |
| 37 | \( 1 - 190 T + p^{3} T^{2} \) |
| 41 | \( 1 + 202 T + p^{3} T^{2} \) |
| 43 | \( 1 - 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 124 T + p^{3} T^{2} \) |
| 53 | \( 1 + 206 T + p^{3} T^{2} \) |
| 59 | \( 1 + 128 T + p^{3} T^{2} \) |
| 61 | \( 1 - 190 T + p^{3} T^{2} \) |
| 67 | \( 1 - 204 T + p^{3} T^{2} \) |
| 71 | \( 1 + 440 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 79 | \( 1 + 816 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1412 T + p^{3} T^{2} \) |
| 89 | \( 1 - 214 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1202 T + p^{3} T^{2} \) |
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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
−8.596009101264268901615977740946, −8.052448932088165840640417275772, −7.24922944837136919963292779525, −6.29687729877037352317403189296, −5.37383114228527808554612771293, −4.50305967378684014932242907157, −3.64094120651138828383101231820, −2.49286487754356437406755043437, −1.33717700198834086459302718927, 0,
1.33717700198834086459302718927, 2.49286487754356437406755043437, 3.64094120651138828383101231820, 4.50305967378684014932242907157, 5.37383114228527808554612771293, 6.29687729877037352317403189296, 7.24922944837136919963292779525, 8.052448932088165840640417275772, 8.596009101264268901615977740946
