| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 2·11-s + 12-s + 2·13-s + 14-s + 15-s + 16-s + 5·17-s − 18-s + 20-s − 21-s − 2·22-s + 6·23-s − 24-s + 25-s − 2·26-s + 27-s − 28-s − 4·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s − 0.426·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.708140489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.708140489\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−14.08972711301416, −13.59503993698868, −13.03223928104663, −12.63622921947667, −12.02903601619500, −11.46879158385580, −11.01841143146986, −10.23958315002889, −10.07422017324100, −9.404631128250405, −9.022844989537924, −8.634655654494866, −7.894191971370529, −7.595257590170853, −6.755997757120305, −6.526906562353082, −5.869352292131715, −5.187222347993562, −4.619917277177392, −3.588723766564356, −3.331205739867530, −2.757113292148354, −1.758998887688138, −1.436629394533007, −0.6000482778043212,
0.6000482778043212, 1.436629394533007, 1.758998887688138, 2.757113292148354, 3.331205739867530, 3.588723766564356, 4.619917277177392, 5.187222347993562, 5.869352292131715, 6.526906562353082, 6.755997757120305, 7.595257590170853, 7.894191971370529, 8.634655654494866, 9.022844989537924, 9.404631128250405, 10.07422017324100, 10.23958315002889, 11.01841143146986, 11.46879158385580, 12.02903601619500, 12.63622921947667, 13.03223928104663, 13.59503993698868, 14.08972711301416
