L(s) = 1 | + 2.66·2-s + 5.12·4-s − 1.45·5-s + 7-s + 8.33·8-s − 3.88·10-s − 1.54·11-s + 5.88·13-s + 2.66·14-s + 12.0·16-s − 6.79·17-s − 6.24·19-s − 7.45·20-s − 4.12·22-s + 2.90·23-s − 2.88·25-s + 15.7·26-s + 5.12·28-s + 3.88·29-s + 2·31-s + 15.3·32-s − 18.1·34-s − 1.45·35-s + 5·37-s − 16.6·38-s − 12.1·40-s − 2.24·41-s + ⋯ |
L(s) = 1 | + 1.88·2-s + 2.56·4-s − 0.650·5-s + 0.377·7-s + 2.94·8-s − 1.22·10-s − 0.465·11-s + 1.63·13-s + 0.713·14-s + 3.00·16-s − 1.64·17-s − 1.43·19-s − 1.66·20-s − 0.879·22-s + 0.606·23-s − 0.576·25-s + 3.07·26-s + 0.968·28-s + 0.721·29-s + 0.359·31-s + 2.71·32-s − 3.10·34-s − 0.245·35-s + 0.821·37-s − 2.70·38-s − 1.91·40-s − 0.351·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.117880930\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.117880930\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 2.24T + 41T^{2} \) |
| 43 | \( 1 + 7.13T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 + 9.79T + 53T^{2} \) |
| 59 | \( 1 + 4.67T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 - 3.36T + 67T^{2} \) |
| 71 | \( 1 - 1.36T + 71T^{2} \) |
| 73 | \( 1 + 1.88T + 73T^{2} \) |
| 79 | \( 1 - 3.36T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 0.793T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−11.17963093200107522832920889819, −10.50520979341109717414166055268, −8.675807900136106353411101632154, −7.88560486120336934818194137283, −6.62036596123719874308845812674, −6.17642581414690064612506196135, −4.82726169756478738766241634704, −4.24350905981333780661521834035, −3.25618973559653280192275619474, −1.97327481307346946432861525220,
1.97327481307346946432861525220, 3.25618973559653280192275619474, 4.24350905981333780661521834035, 4.82726169756478738766241634704, 6.17642581414690064612506196135, 6.62036596123719874308845812674, 7.88560486120336934818194137283, 8.675807900136106353411101632154, 10.50520979341109717414166055268, 11.17963093200107522832920889819