L(s) = 1 | − 2-s − 1.48·3-s + 4-s − 5-s + 1.48·6-s + 4.09·7-s − 8-s − 0.805·9-s + 10-s − 4.92·11-s − 1.48·12-s − 4.05·13-s − 4.09·14-s + 1.48·15-s + 16-s − 4.30·17-s + 0.805·18-s − 0.498·19-s − 20-s − 6.07·21-s + 4.92·22-s + 8.68·23-s + 1.48·24-s + 25-s + 4.05·26-s + 5.63·27-s + 4.09·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.855·3-s + 0.5·4-s − 0.447·5-s + 0.604·6-s + 1.54·7-s − 0.353·8-s − 0.268·9-s + 0.316·10-s − 1.48·11-s − 0.427·12-s − 1.12·13-s − 1.09·14-s + 0.382·15-s + 0.250·16-s − 1.04·17-s + 0.189·18-s − 0.114·19-s − 0.223·20-s − 1.32·21-s + 1.05·22-s + 1.81·23-s + 0.302·24-s + 0.200·25-s + 0.795·26-s + 1.08·27-s + 0.774·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 1.48T + 3T^{2} \) |
| 7 | \( 1 - 4.09T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 + 0.498T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 5.30T + 31T^{2} \) |
| 37 | \( 1 - 4.05T + 37T^{2} \) |
| 41 | \( 1 - 0.930T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 - 8.99T + 47T^{2} \) |
| 53 | \( 1 + 9.36T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 4.71T + 61T^{2} \) |
| 67 | \( 1 + 0.367T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 + 8.25T + 89T^{2} \) |
| 97 | \( 1 + 8.00T + 97T^{2} \) |
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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
−7.64024270285914848441344667566, −7.33722811302699035830776302907, −6.37127598975645765188196495880, −5.43522232398330155612444489277, −4.92845841680684726340867820155, −4.44305926669696444978081021573, −2.82387375375530215326451414847, −2.31987762747292802539578656304, −1.00331575634137464545496695052, 0,
1.00331575634137464545496695052, 2.31987762747292802539578656304, 2.82387375375530215326451414847, 4.44305926669696444978081021573, 4.92845841680684726340867820155, 5.43522232398330155612444489277, 6.37127598975645765188196495880, 7.33722811302699035830776302907, 7.64024270285914848441344667566