| L(s) = 1 | + 1.43·2-s + 0.0711·4-s + 2.31·5-s − 4.44·7-s − 2.77·8-s + 3.33·10-s + 4.52·11-s + 1.17·13-s − 6.39·14-s − 4.13·16-s + 4.86·19-s + 0.164·20-s + 6.51·22-s + 0.625·23-s + 0.354·25-s + 1.69·26-s − 0.316·28-s + 1.51·29-s + 8.73·31-s − 0.402·32-s − 10.2·35-s + 8.79·37-s + 7.00·38-s − 6.42·40-s + 0.464·41-s − 1.51·43-s + 0.321·44-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.0355·4-s + 1.03·5-s − 1.67·7-s − 0.981·8-s + 1.05·10-s + 1.36·11-s + 0.325·13-s − 1.70·14-s − 1.03·16-s + 1.11·19-s + 0.0368·20-s + 1.38·22-s + 0.130·23-s + 0.0708·25-s + 0.331·26-s − 0.0597·28-s + 0.280·29-s + 1.56·31-s − 0.0711·32-s − 1.73·35-s + 1.44·37-s + 1.13·38-s − 1.01·40-s + 0.0726·41-s − 0.230·43-s + 0.0485·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.941808847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.941808847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 5 | \( 1 - 2.31T + 5T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 19 | \( 1 - 4.86T + 19T^{2} \) |
| 23 | \( 1 - 0.625T + 23T^{2} \) |
| 29 | \( 1 - 1.51T + 29T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 - 8.79T + 37T^{2} \) |
| 41 | \( 1 - 0.464T + 41T^{2} \) |
| 43 | \( 1 + 1.51T + 43T^{2} \) |
| 47 | \( 1 + 6.01T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.55T + 61T^{2} \) |
| 67 | \( 1 + 3.71T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 4.45T + 83T^{2} \) |
| 89 | \( 1 + 1.54T + 89T^{2} \) |
| 97 | \( 1 + 3.83T + 97T^{2} \) |
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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
−9.270066241127427157154657671879, −8.207253873484113656417330740780, −6.76270642153978991895490456561, −6.40442672341066087242562672280, −5.88928754375053272803344114724, −4.99879939435456387372387971405, −3.96789378043022976715306135869, −3.32631415256266792416812456942, −2.52148021056827601507109655046, −0.954509145927745915830424954681,
0.954509145927745915830424954681, 2.52148021056827601507109655046, 3.32631415256266792416812456942, 3.96789378043022976715306135869, 4.99879939435456387372387971405, 5.88928754375053272803344114724, 6.40442672341066087242562672280, 6.76270642153978991895490456561, 8.207253873484113656417330740780, 9.270066241127427157154657671879
