| L(s) = 1 | − 2-s + 4-s − 5-s − 0.618·7-s − 8-s + 10-s + 2.85·11-s − 7.09·13-s + 0.618·14-s + 16-s − 6.09·17-s + 1.85·19-s − 20-s − 2.85·22-s − 23-s + 25-s + 7.09·26-s − 0.618·28-s + 9.23·29-s + 9.09·31-s − 32-s + 6.09·34-s + 0.618·35-s + 6.47·37-s − 1.85·38-s + 40-s − 3.32·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.233·7-s − 0.353·8-s + 0.316·10-s + 0.860·11-s − 1.96·13-s + 0.165·14-s + 0.250·16-s − 1.47·17-s + 0.425·19-s − 0.223·20-s − 0.608·22-s − 0.208·23-s + 0.200·25-s + 1.39·26-s − 0.116·28-s + 1.71·29-s + 1.63·31-s − 0.176·32-s + 1.04·34-s + 0.104·35-s + 1.06·37-s − 0.300·38-s + 0.158·40-s − 0.519·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8933470998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8933470998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 7.09T + 13T^{2} \) |
| 17 | \( 1 + 6.09T + 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 3.70T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 9.32T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 4.09T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−9.223470128264673063072497875000, −8.326617505517797241380144977711, −7.67707462950570956460978681126, −6.74188919941959306768373955162, −6.41471522684785427656491994896, −4.92810056619173249586244672859, −4.34509925821010978980810211039, −3.01374878089399841621000142033, −2.20303926815927969478566357017, −0.67445417495635941704012673035,
0.67445417495635941704012673035, 2.20303926815927969478566357017, 3.01374878089399841621000142033, 4.34509925821010978980810211039, 4.92810056619173249586244672859, 6.41471522684785427656491994896, 6.74188919941959306768373955162, 7.67707462950570956460978681126, 8.326617505517797241380144977711, 9.223470128264673063072497875000
