| L(s) = 1 | − 5·5-s − 24.0·7-s + 2.95·11-s − 22.1·13-s + 76.0·17-s + 72.1·19-s − 176.·23-s + 25·25-s + 42.5·29-s + 327.·31-s + 120.·35-s + 182.·37-s + 154.·41-s − 173.·43-s − 338.·47-s + 236.·49-s − 26.5·53-s − 14.7·55-s + 391.·59-s − 191.·61-s + 110.·65-s + 507.·67-s + 576.·71-s − 390.·73-s − 71.2·77-s − 1.22e3·79-s − 247.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.30·7-s + 0.0811·11-s − 0.473·13-s + 1.08·17-s + 0.870·19-s − 1.59·23-s + 0.200·25-s + 0.272·29-s + 1.89·31-s + 0.581·35-s + 0.809·37-s + 0.588·41-s − 0.615·43-s − 1.04·47-s + 0.690·49-s − 0.0688·53-s − 0.0362·55-s + 0.864·59-s − 0.401·61-s + 0.211·65-s + 0.925·67-s + 0.963·71-s − 0.625·73-s − 0.105·77-s − 1.73·79-s − 0.327·83-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 | \( 1 + 5T \) |
| good | 7 | \( 1 + 24.0T + 343T^{2} \) |
| 11 | \( 1 - 2.95T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 327.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 173.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 26.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 391.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 191.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 507.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 576.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 390.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 247.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 959.T + 9.12e5T^{2} \) |
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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
−8.146649469004933684224130183640, −7.68601623587707586930118913229, −6.67595106048498905148214056269, −6.11327339777450725188285239814, −5.16130447365789792495573832442, −4.12449522903152772273199369256, −3.32144502962487090008675734252, −2.57351058249551052627605258341, −1.05595781890755154060583478068, 0,
1.05595781890755154060583478068, 2.57351058249551052627605258341, 3.32144502962487090008675734252, 4.12449522903152772273199369256, 5.16130447365789792495573832442, 6.11327339777450725188285239814, 6.67595106048498905148214056269, 7.68601623587707586930118913229, 8.146649469004933684224130183640
