| L(s) = 1 | + 2.28·2-s + 0.0269·3-s − 2.77·4-s − 5·5-s + 0.0616·6-s − 20.0·7-s − 24.6·8-s − 26.9·9-s − 11.4·10-s − 11·11-s − 0.0748·12-s + 58.5·13-s − 45.8·14-s − 0.134·15-s − 34.0·16-s − 33.3·17-s − 61.7·18-s − 19·19-s + 13.8·20-s − 0.540·21-s − 25.1·22-s − 167.·23-s − 0.664·24-s + 25·25-s + 133.·26-s − 1.45·27-s + 55.7·28-s + ⋯ |
| L(s) = 1 | + 0.808·2-s + 0.00518·3-s − 0.347·4-s − 0.447·5-s + 0.00419·6-s − 1.08·7-s − 1.08·8-s − 0.999·9-s − 0.361·10-s − 0.301·11-s − 0.00180·12-s + 1.24·13-s − 0.875·14-s − 0.00232·15-s − 0.532·16-s − 0.476·17-s − 0.808·18-s − 0.229·19-s + 0.155·20-s − 0.00562·21-s − 0.243·22-s − 1.51·23-s − 0.00564·24-s + 0.200·25-s + 1.00·26-s − 0.0103·27-s + 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9900232542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9900232542\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 2.28T + 8T^{2} \) |
| 3 | \( 1 - 0.0269T + 27T^{2} \) |
| 7 | \( 1 + 20.0T + 343T^{2} \) |
| 13 | \( 1 - 58.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.3T + 4.91e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 545.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 190.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 738.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 661.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 70.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 149.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 533.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 157.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 84.9T + 4.93e5T^{2} \) |
| 83 | \( 1 - 971.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 89.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 932.T + 9.12e5T^{2} \) |
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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
−9.507550643663596353517382696631, −8.595819413228911888761423609156, −8.124488896667146001568968516508, −6.61161391700632615178859736105, −6.07087715147056467126180467276, −5.26672626457780091622306832112, −4.01502783499705939256111039855, −3.52357797611837159377673885357, −2.49913820922541470278500256454, −0.44053977575809146838826925766,
0.44053977575809146838826925766, 2.49913820922541470278500256454, 3.52357797611837159377673885357, 4.01502783499705939256111039855, 5.26672626457780091622306832112, 6.07087715147056467126180467276, 6.61161391700632615178859736105, 8.124488896667146001568968516508, 8.595819413228911888761423609156, 9.507550643663596353517382696631
