| L(s) = 1 | − 2.14·2-s + 2.38·3-s − 3.39·4-s − 5.12·6-s + 24.4·8-s − 21.2·9-s − 15.0·11-s − 8.10·12-s + 16.0·13-s − 25.3·16-s + 55.2·17-s + 45.7·18-s − 81.7·19-s + 32.2·22-s + 108.·23-s + 58.4·24-s − 34.3·26-s − 115.·27-s + 112.·29-s + 90.6·31-s − 141.·32-s − 35.9·33-s − 118.·34-s + 72.2·36-s − 181.·37-s + 175.·38-s + 38.2·39-s + ⋯ |
| L(s) = 1 | − 0.758·2-s + 0.459·3-s − 0.424·4-s − 0.348·6-s + 1.08·8-s − 0.788·9-s − 0.412·11-s − 0.194·12-s + 0.341·13-s − 0.396·16-s + 0.788·17-s + 0.598·18-s − 0.987·19-s + 0.312·22-s + 0.986·23-s + 0.496·24-s − 0.259·26-s − 0.822·27-s + 0.723·29-s + 0.525·31-s − 0.780·32-s − 0.189·33-s − 0.598·34-s + 0.334·36-s − 0.805·37-s + 0.749·38-s + 0.157·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.14T + 8T^{2} \) |
| 3 | \( 1 - 2.38T + 27T^{2} \) |
| 11 | \( 1 + 15.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 55.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 90.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 51.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 198.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 136.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 719.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 926.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 175.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 875.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 737.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 143.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 71.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.46e3T + 9.12e5T^{2} \) |
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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
−8.956302312294389605518314057584, −8.208399617988773059596672244112, −7.72914252478063596096317511019, −6.58340750030465887911087965385, −5.52002901042727589370049999014, −4.63297752476917728913770490107, −3.53500921872156297062096670953, −2.50874677724628529602172429628, −1.18106140746089649713731128247, 0,
1.18106140746089649713731128247, 2.50874677724628529602172429628, 3.53500921872156297062096670953, 4.63297752476917728913770490107, 5.52002901042727589370049999014, 6.58340750030465887911087965385, 7.72914252478063596096317511019, 8.208399617988773059596672244112, 8.956302312294389605518314057584
