L(s) = 1 | − 2-s + 2.93·3-s + 4-s − 1.21·5-s − 2.93·6-s + 3.36·7-s − 8-s + 5.62·9-s + 1.21·10-s − 3.98·11-s + 2.93·12-s + 1.52·13-s − 3.36·14-s − 3.56·15-s + 16-s − 3.62·17-s − 5.62·18-s + 2.25·19-s − 1.21·20-s + 9.87·21-s + 3.98·22-s + 23-s − 2.93·24-s − 3.52·25-s − 1.52·26-s + 7.69·27-s + 3.36·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.69·3-s + 0.5·4-s − 0.543·5-s − 1.19·6-s + 1.27·7-s − 0.353·8-s + 1.87·9-s + 0.384·10-s − 1.20·11-s + 0.847·12-s + 0.424·13-s − 0.898·14-s − 0.921·15-s + 0.250·16-s − 0.879·17-s − 1.32·18-s + 0.516·19-s − 0.271·20-s + 2.15·21-s + 0.849·22-s + 0.208·23-s − 0.599·24-s − 0.704·25-s − 0.300·26-s + 1.48·27-s + 0.635·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.842296439\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.842296439\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 + 1.21T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 - 1.52T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 - 2.25T + 19T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 - 1.10T + 43T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 9.51T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 + 2.15T + 67T^{2} \) |
| 71 | \( 1 - 8.98T + 71T^{2} \) |
| 73 | \( 1 - 2.80T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 9.09T + 83T^{2} \) |
| 89 | \( 1 - 18.8T + 89T^{2} \) |
| 97 | \( 1 + 7.05T + 97T^{2} \) |
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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
−8.131976674037306598813437134648, −7.64071677242377123188654860406, −7.27629197588290961165457072874, −6.09051351959613096236920603578, −5.00218849654434713809366244999, −4.27486037512134069912765039562, −3.47478851102571375578159413506, −2.50532167548794590313590469064, −2.09034797399966847912597995890, −0.923712022476154376732299255602,
0.923712022476154376732299255602, 2.09034797399966847912597995890, 2.50532167548794590313590469064, 3.47478851102571375578159413506, 4.27486037512134069912765039562, 5.00218849654434713809366244999, 6.09051351959613096236920603578, 7.27629197588290961165457072874, 7.64071677242377123188654860406, 8.131976674037306598813437134648