L(s) = 1 | − 2-s + 1.53·3-s + 4-s − 0.384·5-s − 1.53·6-s − 3.26·7-s − 8-s − 0.652·9-s + 0.384·10-s − 2.58·11-s + 1.53·12-s − 1.36·13-s + 3.26·14-s − 0.589·15-s + 16-s + 3.33·17-s + 0.652·18-s + 5.98·19-s − 0.384·20-s − 5.00·21-s + 2.58·22-s + 7.54·23-s − 1.53·24-s − 4.85·25-s + 1.36·26-s − 5.59·27-s − 3.26·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.884·3-s + 0.5·4-s − 0.172·5-s − 0.625·6-s − 1.23·7-s − 0.353·8-s − 0.217·9-s + 0.121·10-s − 0.779·11-s + 0.442·12-s − 0.378·13-s + 0.872·14-s − 0.152·15-s + 0.250·16-s + 0.808·17-s + 0.153·18-s + 1.37·19-s − 0.0860·20-s − 1.09·21-s + 0.551·22-s + 1.57·23-s − 0.312·24-s − 0.970·25-s + 0.267·26-s − 1.07·27-s − 0.616·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.224271526\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.224271526\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 5 | \( 1 + 0.384T + 5T^{2} \) |
| 7 | \( 1 + 3.26T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 - 5.98T + 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 + 4.33T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 + 8.67T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 7.01T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 + 0.0111T + 83T^{2} \) |
| 89 | \( 1 + 0.953T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.956904387552916339088592585050, −7.960080794865412405902196120699, −7.62278742513875183122478611992, −6.79270718853099949191069736123, −5.83828804795423434043653002351, −5.05457217745048583878805444010, −3.48425798936028994544415541333, −3.13040105567542438981221850459, −2.27837501298876929263722725891, −0.70567350604814230070825810532,
0.70567350604814230070825810532, 2.27837501298876929263722725891, 3.13040105567542438981221850459, 3.48425798936028994544415541333, 5.05457217745048583878805444010, 5.83828804795423434043653002351, 6.79270718853099949191069736123, 7.62278742513875183122478611992, 7.960080794865412405902196120699, 8.956904387552916339088592585050