| L(s) = 1 | + 1.19·2-s − 0.568·4-s + 0.744·5-s + 4.07·7-s − 3.07·8-s + 0.890·10-s − 11-s − 1.12·13-s + 4.87·14-s − 2.54·16-s − 5.26·17-s − 1.09·19-s − 0.422·20-s − 1.19·22-s + 0.724·23-s − 4.44·25-s − 1.34·26-s − 2.31·28-s − 0.375·29-s + 9.00·31-s + 3.10·32-s − 6.29·34-s + 3.03·35-s + 11.2·37-s − 1.30·38-s − 2.28·40-s − 6.79·41-s + ⋯ |
| L(s) = 1 | + 0.846·2-s − 0.284·4-s + 0.332·5-s + 1.54·7-s − 1.08·8-s + 0.281·10-s − 0.301·11-s − 0.311·13-s + 1.30·14-s − 0.635·16-s − 1.27·17-s − 0.250·19-s − 0.0945·20-s − 0.255·22-s + 0.150·23-s − 0.889·25-s − 0.263·26-s − 0.437·28-s − 0.0697·29-s + 1.61·31-s + 0.549·32-s − 1.08·34-s + 0.512·35-s + 1.85·37-s − 0.212·38-s − 0.361·40-s − 1.06·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.027807310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.027807310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 5 | \( 1 - 0.744T + 5T^{2} \) |
| 7 | \( 1 - 4.07T + 7T^{2} \) |
| 13 | \( 1 + 1.12T + 13T^{2} \) |
| 17 | \( 1 + 5.26T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 - 0.724T + 23T^{2} \) |
| 29 | \( 1 + 0.375T + 29T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 6.24T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 9.85T + 61T^{2} \) |
| 71 | \( 1 + 0.951T + 71T^{2} \) |
| 73 | \( 1 + 2.36T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 - 4.53T + 83T^{2} \) |
| 89 | \( 1 - 5.18T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.982139225179923117183964942927, −7.34593351090375420780145033263, −6.26137148135625683060255368628, −5.80553815895849576399267679379, −4.89364186058589418637394531186, −4.54958087784054279523371038741, −3.93518799770738444493275380359, −2.64341378960741789485968910124, −2.12878451486983552834942190748, −0.789973432241783122494547869578,
0.789973432241783122494547869578, 2.12878451486983552834942190748, 2.64341378960741789485968910124, 3.93518799770738444493275380359, 4.54958087784054279523371038741, 4.89364186058589418637394531186, 5.80553815895849576399267679379, 6.26137148135625683060255368628, 7.34593351090375420780145033263, 7.982139225179923117183964942927
