| L(s) = 1 | + 1.73·3-s − 2.73·5-s − 1.26·7-s + 4.73·11-s − 0.464·13-s − 4.73·15-s − 6.19·17-s − 3.46·19-s − 2.19·21-s + 23-s + 2.46·25-s − 5.19·27-s − 1.53·29-s − 7.73·31-s + 8.19·33-s + 3.46·35-s + 4.19·37-s − 0.803·39-s − 8.46·41-s + 6.92·43-s − 0.803·47-s − 5.39·49-s − 10.7·51-s + 4.92·53-s − 12.9·55-s − 5.99·57-s + 2.53·59-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 1.22·5-s − 0.479·7-s + 1.42·11-s − 0.128·13-s − 1.22·15-s − 1.50·17-s − 0.794·19-s − 0.479·21-s + 0.208·23-s + 0.492·25-s − 1.00·27-s − 0.285·29-s − 1.38·31-s + 1.42·33-s + 0.585·35-s + 0.689·37-s − 0.128·39-s − 1.32·41-s + 1.05·43-s − 0.117·47-s − 0.770·49-s − 1.50·51-s + 0.676·53-s − 1.74·55-s − 0.794·57-s + 0.330·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 + 0.464T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 + 7.73T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 + 8.46T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 + 0.803T + 47T^{2} \) |
| 53 | \( 1 - 4.92T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 6.46T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 4.73T + 97T^{2} \) |
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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
−8.902392897103418166155764373371, −8.534308042722768650807752273208, −7.52157927395237607189808932128, −6.88478406047734086928963328815, −5.98948369187569748728520525295, −4.44028217898655617968492804102, −3.89867540204955331422701762284, −3.10160616384275556303044185155, −1.91543091146839422241320425220, 0,
1.91543091146839422241320425220, 3.10160616384275556303044185155, 3.89867540204955331422701762284, 4.44028217898655617968492804102, 5.98948369187569748728520525295, 6.88478406047734086928963328815, 7.52157927395237607189808932128, 8.534308042722768650807752273208, 8.902392897103418166155764373371
