L(s) = 1 | + 2.14·2-s + 1.40·3-s + 2.60·4-s + 5-s + 3.00·6-s + 5.15·7-s + 1.29·8-s − 1.03·9-s + 2.14·10-s + 0.220·11-s + 3.64·12-s + 3.46·13-s + 11.0·14-s + 1.40·15-s − 2.43·16-s + 4.13·17-s − 2.22·18-s − 2.82·19-s + 2.60·20-s + 7.22·21-s + 0.472·22-s + 1.81·24-s + 25-s + 7.43·26-s − 5.65·27-s + 13.4·28-s − 8.82·29-s + ⋯ |
L(s) = 1 | + 1.51·2-s + 0.808·3-s + 1.30·4-s + 0.447·5-s + 1.22·6-s + 1.94·7-s + 0.457·8-s − 0.345·9-s + 0.678·10-s + 0.0663·11-s + 1.05·12-s + 0.960·13-s + 2.95·14-s + 0.361·15-s − 0.607·16-s + 1.00·17-s − 0.524·18-s − 0.648·19-s + 0.582·20-s + 1.57·21-s + 0.100·22-s + 0.369·24-s + 0.200·25-s + 1.45·26-s − 1.08·27-s + 2.53·28-s − 1.63·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.037517344\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.037517344\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 7 | \( 1 - 5.15T + 7T^{2} \) |
| 11 | \( 1 - 0.220T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 + 9.62T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 - 0.699T + 41T^{2} \) |
| 43 | \( 1 - 2.36T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 8.75T + 67T^{2} \) |
| 71 | \( 1 - 7.24T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 - 2.31T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.808299907308633666041388569088, −7.975013336905073974027000345490, −7.43056808601143998027591252182, −6.17976483556974821724185373553, −5.51624440004693676807054983036, −5.01382245619227866553013997497, −3.96248651331527078641927418854, −3.44461421732376298599137718157, −2.23828217481091357610956279748, −1.64678415728943658953267783698,
1.64678415728943658953267783698, 2.23828217481091357610956279748, 3.44461421732376298599137718157, 3.96248651331527078641927418854, 5.01382245619227866553013997497, 5.51624440004693676807054983036, 6.17976483556974821724185373553, 7.43056808601143998027591252182, 7.975013336905073974027000345490, 8.808299907308633666041388569088