| L(s) = 1 | − 5.00·2-s − 9.04·3-s + 17.0·4-s − 15.8·5-s + 45.2·6-s + 23.4·7-s − 45.3·8-s + 54.7·9-s + 79.4·10-s − 154.·12-s − 13·13-s − 117.·14-s + 143.·15-s + 90.4·16-s + 114.·17-s − 274.·18-s + 112.·19-s − 270.·20-s − 212.·21-s − 137.·23-s + 409.·24-s + 126.·25-s + 65.0·26-s − 251.·27-s + 400.·28-s + 147.·29-s − 718.·30-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 1.74·3-s + 2.13·4-s − 1.41·5-s + 3.07·6-s + 1.26·7-s − 2.00·8-s + 2.02·9-s + 2.51·10-s − 3.71·12-s − 0.277·13-s − 2.24·14-s + 2.47·15-s + 1.41·16-s + 1.62·17-s − 3.59·18-s + 1.36·19-s − 3.02·20-s − 2.20·21-s − 1.25·23-s + 3.48·24-s + 1.01·25-s + 0.490·26-s − 1.79·27-s + 2.70·28-s + 0.945·29-s − 4.37·30-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 | \( 1 + 5.00T + 8T^{2} \) |
| 3 | \( 1 + 9.04T + 27T^{2} \) |
| 5 | \( 1 + 15.8T + 125T^{2} \) |
| 7 | \( 1 - 23.4T + 343T^{2} \) |
| 17 | \( 1 - 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 147.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 4.37T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 356.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 510.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 111.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 45.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 680.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 397.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 825.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 703.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 139.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 340.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 873.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 295.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 87.2T + 9.12e5T^{2} \) |
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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
−8.285628588486091728547430203193, −7.982960925133747505576631198760, −7.31610486527990094427268747859, −6.58952964893385376668179071294, −5.41827298311220789466454549375, −4.77428362752342902665096479371, −3.47293532103427450450402712588, −1.61384079982863046896343760317, −0.881605215676777523575853687817, 0,
0.881605215676777523575853687817, 1.61384079982863046896343760317, 3.47293532103427450450402712588, 4.77428362752342902665096479371, 5.41827298311220789466454549375, 6.58952964893385376668179071294, 7.31610486527990094427268747859, 7.982960925133747505576631198760, 8.285628588486091728547430203193
