| L(s) = 1 | + 2.23·3-s + 4.47·7-s + 2.00·9-s + 2.23·11-s + 4·13-s − 7·17-s − 6.70·19-s + 10.0·21-s − 4.47·23-s − 2.23·27-s + 4.47·31-s + 5.00·33-s + 2·37-s + 8.94·39-s + 5·41-s − 8.94·47-s + 13.0·49-s − 15.6·51-s + 6·53-s − 15.0·57-s + 8.94·59-s + 10·61-s + 8.94·63-s − 2.23·67-s − 10.0·69-s − 8.94·71-s − 9·73-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 1.69·7-s + 0.666·9-s + 0.674·11-s + 1.10·13-s − 1.69·17-s − 1.53·19-s + 2.18·21-s − 0.932·23-s − 0.430·27-s + 0.803·31-s + 0.870·33-s + 0.328·37-s + 1.43·39-s + 0.780·41-s − 1.30·47-s + 1.85·49-s − 2.19·51-s + 0.824·53-s − 1.98·57-s + 1.16·59-s + 1.28·61-s + 1.12·63-s − 0.273·67-s − 1.20·69-s − 1.06·71-s − 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.744353534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.744353534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.24993019668767040986526642586, −9.014783488310806066676906286092, −8.465430342484352186093050379282, −8.167905739342344370023067006366, −6.94760004129955496294405041358, −5.94426147354046117803981663122, −4.43230042203715345727770067725, −4.01699601747083075472287508810, −2.43843119291001336407519068539, −1.64894313034439815950593792217,
1.64894313034439815950593792217, 2.43843119291001336407519068539, 4.01699601747083075472287508810, 4.43230042203715345727770067725, 5.94426147354046117803981663122, 6.94760004129955496294405041358, 8.167905739342344370023067006366, 8.465430342484352186093050379282, 9.014783488310806066676906286092, 10.24993019668767040986526642586
