| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 21.6·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 43.3·10-s − 7.67·11-s − 12·12-s + 63.7·13-s + 14·14-s − 64.9·15-s + 16·16-s + 9.23·17-s − 18·18-s − 145.·19-s + 86.6·20-s + 21·21-s + 15.3·22-s − 23·23-s + 24·24-s + 344.·25-s − 127.·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.93·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.36·10-s − 0.210·11-s − 0.288·12-s + 1.36·13-s + 0.267·14-s − 1.11·15-s + 0.250·16-s + 0.131·17-s − 0.235·18-s − 1.75·19-s + 0.968·20-s + 0.218·21-s + 0.148·22-s − 0.208·23-s + 0.204·24-s + 2.75·25-s − 0.961·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.840597247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.840597247\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 21.6T + 125T^{2} \) |
| 11 | \( 1 + 7.67T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.23T + 4.91e3T^{2} \) |
| 19 | \( 1 + 145.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 292.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 121.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 91.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 10.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 10.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 283.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 308.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 139.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 888.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 778.T + 9.12e5T^{2} \) |
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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
−9.754369819661060717565596101409, −8.906644918370238490613818272865, −8.248738600889791908807505023376, −6.68246487925562032187436345112, −6.30314640881976641826421130692, −5.71040498285041390409617054210, −4.49046117501266821193791350457, −2.85971769025966563316204199974, −1.85869403075341103525137614930, −0.867055540327032214383416462821,
0.867055540327032214383416462821, 1.85869403075341103525137614930, 2.85971769025966563316204199974, 4.49046117501266821193791350457, 5.71040498285041390409617054210, 6.30314640881976641826421130692, 6.68246487925562032187436345112, 8.248738600889791908807505023376, 8.906644918370238490613818272865, 9.754369819661060717565596101409
