| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 3.19·7-s + 8-s + 9-s − 10-s + 12-s + 6.75·13-s + 3.19·14-s − 15-s + 16-s − 0.557·17-s + 18-s − 6.41·19-s − 20-s + 3.19·21-s + 2.59·23-s + 24-s + 25-s + 6.75·26-s + 27-s + 3.19·28-s + 4.89·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.20·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s + 1.87·13-s + 0.854·14-s − 0.258·15-s + 0.250·16-s − 0.135·17-s + 0.235·18-s − 1.47·19-s − 0.223·20-s + 0.697·21-s + 0.541·23-s + 0.204·24-s + 0.200·25-s + 1.32·26-s + 0.192·27-s + 0.604·28-s + 0.908·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.386402449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.386402449\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 3.19T + 7T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 + 0.557T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 - 2.59T + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 + 5.73T + 37T^{2} \) |
| 41 | \( 1 + 1.31T + 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 + 5.99T + 47T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 3.92T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 6.35T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 7.63T + 97T^{2} \) |
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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
−8.338051176758676535892770050784, −8.086166787536254960618009055217, −6.92288685239113575269799325548, −6.39849221115221215345443697333, −5.38695492397441688142152045062, −4.54663916375618092336784721960, −3.98456098404068093739145477486, −3.17612640118119707047025608070, −2.07836983073051770458683916511, −1.20105954923881766848373187949,
1.20105954923881766848373187949, 2.07836983073051770458683916511, 3.17612640118119707047025608070, 3.98456098404068093739145477486, 4.54663916375618092336784721960, 5.38695492397441688142152045062, 6.39849221115221215345443697333, 6.92288685239113575269799325548, 8.086166787536254960618009055217, 8.338051176758676535892770050784
