| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 2.35·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 4.70·10-s − 65.1·11-s − 12·12-s − 31.4·13-s + 14·14-s + 7.05·15-s + 16·16-s − 43.0·17-s − 18·18-s − 19·19-s − 9.40·20-s + 21·21-s + 130.·22-s + 92.8·23-s + 24·24-s − 119.·25-s + 62.9·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.210·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.148·10-s − 1.78·11-s − 0.288·12-s − 0.671·13-s + 0.267·14-s + 0.121·15-s + 0.250·16-s − 0.614·17-s − 0.235·18-s − 0.229·19-s − 0.105·20-s + 0.218·21-s + 1.26·22-s + 0.841·23-s + 0.204·24-s − 0.955·25-s + 0.474·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3670177987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3670177987\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 + 2.35T + 125T^{2} \) |
| 11 | \( 1 + 65.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 92.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 95.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.04T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 54.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 220.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 175.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 383.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 431.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 989.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 456.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 92.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 620.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 519.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 166.T + 9.12e5T^{2} \) |
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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
−10.07975305188265469799897379704, −9.095486358935142479205118957160, −8.075587467686266991987323661806, −7.41034129168171869887621587404, −6.52776980428650501183835231446, −5.49714141096704188179023152033, −4.65051666470963939744322995374, −3.13202129984313756749843177126, −2.06926443399384873153478546466, −0.36498004466784405916452090718,
0.36498004466784405916452090718, 2.06926443399384873153478546466, 3.13202129984313756749843177126, 4.65051666470963939744322995374, 5.49714141096704188179023152033, 6.52776980428650501183835231446, 7.41034129168171869887621587404, 8.075587467686266991987323661806, 9.095486358935142479205118957160, 10.07975305188265469799897379704
