| L(s) = 1 | + 2.09·2-s + 7.98·3-s − 3.60·4-s + 17.0·5-s + 16.7·6-s − 31.2·7-s − 24.3·8-s + 36.7·9-s + 35.8·10-s − 28.7·12-s − 13·13-s − 65.6·14-s + 136.·15-s − 22.2·16-s + 70.8·17-s + 77.0·18-s − 144.·19-s − 61.5·20-s − 249.·21-s − 140.·23-s − 194.·24-s + 167.·25-s − 27.2·26-s + 77.5·27-s + 112.·28-s − 267.·29-s + 286.·30-s + ⋯ |
| L(s) = 1 | + 0.741·2-s + 1.53·3-s − 0.450·4-s + 1.52·5-s + 1.13·6-s − 1.68·7-s − 1.07·8-s + 1.35·9-s + 1.13·10-s − 0.691·12-s − 0.277·13-s − 1.25·14-s + 2.34·15-s − 0.347·16-s + 1.01·17-s + 1.00·18-s − 1.74·19-s − 0.688·20-s − 2.59·21-s − 1.26·23-s − 1.65·24-s + 1.33·25-s − 0.205·26-s + 0.552·27-s + 0.760·28-s − 1.71·29-s + 1.74·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 2.09T + 8T^{2} \) |
| 3 | \( 1 - 7.98T + 27T^{2} \) |
| 5 | \( 1 - 17.0T + 125T^{2} \) |
| 7 | \( 1 + 31.2T + 343T^{2} \) |
| 17 | \( 1 - 70.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 3.90T + 5.06e4T^{2} \) |
| 41 | \( 1 + 166.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 267.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 477.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 388.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 161.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 934.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 298.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 983.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 409.T + 9.12e5T^{2} \) |
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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
−9.011665261615628373149941321035, −8.024944150427321180908000557054, −6.89007388519184013351372991930, −6.01254678759608381665747856190, −5.55578079968082815547133221684, −4.07426999519059668390815082497, −3.51654844289836133796376374684, −2.62987263354237397162124827144, −1.92849768645959236309805844837, 0,
1.92849768645959236309805844837, 2.62987263354237397162124827144, 3.51654844289836133796376374684, 4.07426999519059668390815082497, 5.55578079968082815547133221684, 6.01254678759608381665747856190, 6.89007388519184013351372991930, 8.024944150427321180908000557054, 9.011665261615628373149941321035
