L(s) = 1 | − 2-s + 0.732·3-s + 4-s + 5-s − 0.732·6-s − 4.73·7-s − 8-s − 2.46·9-s − 10-s − 5.46·11-s + 0.732·12-s − 5.46·13-s + 4.73·14-s + 0.732·15-s + 16-s + 5.46·17-s + 2.46·18-s + 6.19·19-s + 20-s − 3.46·21-s + 5.46·22-s − 8·23-s − 0.732·24-s + 25-s + 5.46·26-s − 4·27-s − 4.73·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.422·3-s + 0.5·4-s + 0.447·5-s − 0.298·6-s − 1.78·7-s − 0.353·8-s − 0.821·9-s − 0.316·10-s − 1.64·11-s + 0.211·12-s − 1.51·13-s + 1.26·14-s + 0.189·15-s + 0.250·16-s + 1.32·17-s + 0.580·18-s + 1.42·19-s + 0.223·20-s − 0.755·21-s + 1.16·22-s − 1.66·23-s − 0.149·24-s + 0.200·25-s + 1.07·26-s − 0.769·27-s − 0.894·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 - 0.732T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 3.66T + 67T^{2} \) |
| 71 | \( 1 - 2.92T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 - 8.73T + 79T^{2} \) |
| 83 | \( 1 + 8.73T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.46326174053223918931304521411, −9.806964005969077029406927965912, −9.470504733169353546690605268090, −8.038043063656137002117730924866, −7.45397717871850861724929250526, −6.13419658720380308842395948086, −5.33494467746224220922225621034, −3.13746761846583704719760510993, −2.61882371334183910411617060775, 0,
2.61882371334183910411617060775, 3.13746761846583704719760510993, 5.33494467746224220922225621034, 6.13419658720380308842395948086, 7.45397717871850861724929250526, 8.038043063656137002117730924866, 9.470504733169353546690605268090, 9.806964005969077029406927965912, 10.46326174053223918931304521411