L(s) = 1 | − 1.79·2-s − 3-s + 1.23·4-s + 3.97·5-s + 1.79·6-s + 3.17·7-s + 1.37·8-s + 9-s − 7.14·10-s − 11-s − 1.23·12-s + 13-s − 5.70·14-s − 3.97·15-s − 4.94·16-s − 7.80·17-s − 1.79·18-s + 7.17·19-s + 4.90·20-s − 3.17·21-s + 1.79·22-s + 3.36·23-s − 1.37·24-s + 10.7·25-s − 1.79·26-s − 27-s + 3.92·28-s + ⋯ |
L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.617·4-s + 1.77·5-s + 0.734·6-s + 1.19·7-s + 0.485·8-s + 0.333·9-s − 2.25·10-s − 0.301·11-s − 0.356·12-s + 0.277·13-s − 1.52·14-s − 1.02·15-s − 1.23·16-s − 1.89·17-s − 0.424·18-s + 1.64·19-s + 1.09·20-s − 0.692·21-s + 0.383·22-s + 0.700·23-s − 0.280·24-s + 2.15·25-s − 0.352·26-s − 0.192·27-s + 0.741·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9083649671\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9083649671\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 17 | \( 1 + 7.80T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 + 7.61T + 29T^{2} \) |
| 31 | \( 1 - 3.15T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 + 4.15T + 43T^{2} \) |
| 47 | \( 1 - 0.660T + 47T^{2} \) |
| 53 | \( 1 - 0.0696T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 - 5.50T + 67T^{2} \) |
| 71 | \( 1 - 2.11T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 0.965T + 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
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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
−10.99048909037123352781874917856, −10.14792802541874649186358472518, −9.358900766699233379624010239385, −8.741733457527757874305414550112, −7.57652671079220205408341653824, −6.62263700049102057003896726119, −5.46530010436867002387178544749, −4.71706306782640570491021449176, −2.22819009930689211436517355110, −1.27714062245721150583364961300,
1.27714062245721150583364961300, 2.22819009930689211436517355110, 4.71706306782640570491021449176, 5.46530010436867002387178544749, 6.62263700049102057003896726119, 7.57652671079220205408341653824, 8.741733457527757874305414550112, 9.358900766699233379624010239385, 10.14792802541874649186358472518, 10.99048909037123352781874917856