| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 17.0·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s − 34.1·10-s − 63.2·11-s + 12·12-s + 22.2·13-s − 14·14-s − 51.2·15-s + 16·16-s + 53.5·17-s + 18·18-s + 40.5·19-s − 68.2·20-s − 21·21-s − 126.·22-s − 23·23-s + 24·24-s + 166.·25-s + 44.5·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.52·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.07·10-s − 1.73·11-s + 0.288·12-s + 0.474·13-s − 0.267·14-s − 0.881·15-s + 0.250·16-s + 0.764·17-s + 0.235·18-s + 0.489·19-s − 0.763·20-s − 0.218·21-s − 1.22·22-s − 0.208·23-s + 0.204·24-s + 1.33·25-s + 0.335·26-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
| \(L(2)\) |
\(\approx\) |
\(2.608623945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.608623945\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 17.0T + 125T^{2} \) |
| 11 | \( 1 + 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.5T + 6.85e3T^{2} \) |
| 29 | \( 1 - 151.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 83.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 61.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 401.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 159.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 438.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 41.8T + 3.00e5T^{2} \) |
| 71 | \( 1 - 479.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 584.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 555.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 815.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 422.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 124.T + 9.12e5T^{2} \) |
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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
−9.770185669640603291658041249388, −8.422133640700525525035537029674, −7.88134174497622436322239150852, −7.34686937631356227384689668152, −6.17058891010139236095460455830, −5.04804285463515128692754060736, −4.20107667372949206684848668697, −3.26955447759405198057712134322, −2.64258951172290104783124126684, −0.74632725986479337817734529072,
0.74632725986479337817734529072, 2.64258951172290104783124126684, 3.26955447759405198057712134322, 4.20107667372949206684848668697, 5.04804285463515128692754060736, 6.17058891010139236095460455830, 7.34686937631356227384689668152, 7.88134174497622436322239150852, 8.422133640700525525035537029674, 9.770185669640603291658041249388
