| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.651314932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.651314932\) |
| \(L(\frac{3}{2})\) |
|
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
−10.19637291050033323681161148912, −9.277831673235694482561676371872, −8.837733563217110914049869799225, −8.081161836172004388182766644402, −6.67870341459062948868991851127, −6.18778233264734739287836744008, −4.84282698238383160068938081295, −3.81023607085141794938246517960, −2.47131177011771384496032213606, −1.67992304783894860665783630265,
1.67992304783894860665783630265, 2.47131177011771384496032213606, 3.81023607085141794938246517960, 4.84282698238383160068938081295, 6.18778233264734739287836744008, 6.67870341459062948868991851127, 8.081161836172004388182766644402, 8.837733563217110914049869799225, 9.277831673235694482561676371872, 10.19637291050033323681161148912
