| L(s) = 1 | − 0.138·2-s − 1.98·4-s + 5-s + 5.07·7-s + 0.551·8-s − 0.138·10-s + 5.07·11-s − 3.67·13-s − 0.703·14-s + 3.88·16-s + 2.60·17-s − 6.74·19-s − 1.98·20-s − 0.703·22-s + 3.21·23-s + 25-s + 0.508·26-s − 10.0·28-s + 29-s − 0.787·31-s − 1.64·32-s − 0.361·34-s + 5.07·35-s − 5.13·37-s + 0.935·38-s + 0.551·40-s + 8.81·41-s + ⋯ |
| L(s) = 1 | − 0.0979·2-s − 0.990·4-s + 0.447·5-s + 1.91·7-s + 0.195·8-s − 0.0438·10-s + 1.53·11-s − 1.01·13-s − 0.188·14-s + 0.971·16-s + 0.632·17-s − 1.54·19-s − 0.442·20-s − 0.150·22-s + 0.669·23-s + 0.200·25-s + 0.0997·26-s − 1.90·28-s + 0.185·29-s − 0.141·31-s − 0.290·32-s − 0.0619·34-s + 0.858·35-s − 0.843·37-s + 0.151·38-s + 0.0872·40-s + 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.797820936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797820936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.138T + 2T^{2} \) |
| 7 | \( 1 - 5.07T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 3.21T + 23T^{2} \) |
| 31 | \( 1 + 0.787T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 - 8.81T + 41T^{2} \) |
| 43 | \( 1 + 2.61T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 + 0.619T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 8.90T + 61T^{2} \) |
| 67 | \( 1 + 0.524T + 67T^{2} \) |
| 71 | \( 1 + 0.195T + 71T^{2} \) |
| 73 | \( 1 - 1.13T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 18.1T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−9.426871246970957944114033861570, −8.892066035825281577797402962901, −8.168467263465557888254231299925, −7.39268037059723080503452374774, −6.26802939850415108535951581094, −5.16015307118862030253278861309, −4.65492507208300376837167956171, −3.82460131639147294173088142176, −2.11165322967482828365917688400, −1.11887033704630432353220121241,
1.11887033704630432353220121241, 2.11165322967482828365917688400, 3.82460131639147294173088142176, 4.65492507208300376837167956171, 5.16015307118862030253278861309, 6.26802939850415108535951581094, 7.39268037059723080503452374774, 8.168467263465557888254231299925, 8.892066035825281577797402962901, 9.426871246970957944114033861570
