| L(s) = 1 | − 1.28·2-s − 0.351·4-s − 5-s + 1.77·7-s + 3.01·8-s + 1.28·10-s + 6.50·11-s − 13-s − 2.28·14-s − 3.17·16-s − 1.38·17-s − 3.06·19-s + 0.351·20-s − 8.34·22-s + 7.40·23-s + 25-s + 1.28·26-s − 0.624·28-s + 5.26·29-s + 2.04·31-s − 1.96·32-s + 1.77·34-s − 1.77·35-s − 5.90·37-s + 3.93·38-s − 3.01·40-s − 2.31·41-s + ⋯ |
| L(s) = 1 | − 0.907·2-s − 0.175·4-s − 0.447·5-s + 0.672·7-s + 1.06·8-s + 0.406·10-s + 1.96·11-s − 0.277·13-s − 0.610·14-s − 0.793·16-s − 0.334·17-s − 0.702·19-s + 0.0785·20-s − 1.77·22-s + 1.54·23-s + 0.200·25-s + 0.251·26-s − 0.118·28-s + 0.977·29-s + 0.367·31-s − 0.347·32-s + 0.304·34-s − 0.300·35-s − 0.970·37-s + 0.637·38-s − 0.477·40-s − 0.361·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045768578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045768578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.28T + 2T^{2} \) |
| 7 | \( 1 - 1.77T + 7T^{2} \) |
| 11 | \( 1 - 6.50T + 11T^{2} \) |
| 17 | \( 1 + 1.38T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 - 7.40T + 23T^{2} \) |
| 29 | \( 1 - 5.26T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 - 4.06T + 43T^{2} \) |
| 47 | \( 1 + 7.90T + 47T^{2} \) |
| 53 | \( 1 + 0.0333T + 53T^{2} \) |
| 59 | \( 1 - 0.913T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 5.36T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 - 7.13T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.080230463203893293406188458921, −8.687632991902365807648439131105, −7.960847732551982946925438924256, −7.00495001507429844029527532681, −6.46937506957045465945301260839, −4.93793201806239718512513747313, −4.43763241035744691755918730863, −3.44445077262988711213923745244, −1.82555956459565343516581936118, −0.866129235666383133979502649053,
0.866129235666383133979502649053, 1.82555956459565343516581936118, 3.44445077262988711213923745244, 4.43763241035744691755918730863, 4.93793201806239718512513747313, 6.46937506957045465945301260839, 7.00495001507429844029527532681, 7.960847732551982946925438924256, 8.687632991902365807648439131105, 9.080230463203893293406188458921
