| L(s) = 1 | − 2-s − 3-s + 4-s + 3.46·5-s + 6-s − 7-s − 8-s + 9-s − 3.46·10-s + 1.46·11-s − 12-s + 3.46·13-s + 14-s − 3.46·15-s + 16-s + 0.535·17-s − 18-s + 19-s + 3.46·20-s + 21-s − 1.46·22-s + 1.46·23-s + 24-s + 6.99·25-s − 3.46·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.54·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.09·10-s + 0.441·11-s − 0.288·12-s + 0.960·13-s + 0.267·14-s − 0.894·15-s + 0.250·16-s + 0.129·17-s − 0.235·18-s + 0.229·19-s + 0.774·20-s + 0.218·21-s − 0.312·22-s + 0.305·23-s + 0.204·24-s + 1.39·25-s − 0.679·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.260384655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260384655\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−10.17308568957520208426375035051, −9.399890545646732337188201848932, −8.975971051255346749188587677149, −7.65129318108697167228257753205, −6.61328027978598423631447362826, −6.01578748847189244384938899816, −5.31147227846833651158549187373, −3.71372870050451330502111760082, −2.25505237872083819241024314164, −1.14003140960896488450907021944,
1.14003140960896488450907021944, 2.25505237872083819241024314164, 3.71372870050451330502111760082, 5.31147227846833651158549187373, 6.01578748847189244384938899816, 6.61328027978598423631447362826, 7.65129318108697167228257753205, 8.975971051255346749188587677149, 9.399890545646732337188201848932, 10.17308568957520208426375035051
