| L(s) = 1 | − 1.67·2-s + 0.794·4-s + 2.50·5-s − 1.08·7-s + 2.01·8-s − 4.18·10-s − 3.03·11-s − 1.49·13-s + 1.81·14-s − 4.95·16-s + 7.35·17-s + 19-s + 1.98·20-s + 5.07·22-s − 0.330·23-s + 1.26·25-s + 2.50·26-s − 0.861·28-s − 6.86·29-s − 5.10·31-s + 4.25·32-s − 12.2·34-s − 2.71·35-s + 1.14·37-s − 1.67·38-s + 5.04·40-s + 3.58·41-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 0.397·4-s + 1.11·5-s − 0.409·7-s + 0.712·8-s − 1.32·10-s − 0.914·11-s − 0.415·13-s + 0.484·14-s − 1.23·16-s + 1.78·17-s + 0.229·19-s + 0.444·20-s + 1.08·22-s − 0.0689·23-s + 0.252·25-s + 0.491·26-s − 0.162·28-s − 1.27·29-s − 0.917·31-s + 0.752·32-s − 2.10·34-s − 0.458·35-s + 0.188·37-s − 0.271·38-s + 0.797·40-s + 0.560·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 - 7.35T + 17T^{2} \) |
| 23 | \( 1 + 0.330T + 23T^{2} \) |
| 29 | \( 1 + 6.86T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 - 3.58T + 41T^{2} \) |
| 43 | \( 1 - 7.36T + 43T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 6.60T + 59T^{2} \) |
| 61 | \( 1 + 6.25T + 61T^{2} \) |
| 67 | \( 1 - 5.12T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 7.28T + 73T^{2} \) |
| 79 | \( 1 - 7.25T + 79T^{2} \) |
| 83 | \( 1 + 8.11T + 83T^{2} \) |
| 89 | \( 1 - 7.14T + 89T^{2} \) |
| 97 | \( 1 + 7.77T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57661493690981638490930354805, −7.16704011394739273969878051889, −5.98559394660756665958802580682, −5.60532872432580044811032995217, −4.87804280914637227814594505905, −3.77337510466780642818146933133, −2.80684808591438954225098118693, −1.99689348683083402662608288379, −1.16913651943548876301507749000, 0,
1.16913651943548876301507749000, 1.99689348683083402662608288379, 2.80684808591438954225098118693, 3.77337510466780642818146933133, 4.87804280914637227814594505905, 5.60532872432580044811032995217, 5.98559394660756665958802580682, 7.16704011394739273969878051889, 7.57661493690981638490930354805
