| L(s) = 1 | − 1.65·3-s − 0.127·5-s − 0.216·7-s − 0.263·9-s + 1.04·11-s − 13-s + 0.211·15-s − 5.96·17-s − 4.70·19-s + 0.358·21-s + 7.09·23-s − 4.98·25-s + 5.39·27-s + 29-s − 0.651·31-s − 1.73·33-s + 0.0277·35-s − 2.23·37-s + 1.65·39-s − 5.08·41-s − 7.06·43-s + 0.0337·45-s + 9.25·47-s − 6.95·49-s + 9.86·51-s + 3.34·53-s − 0.133·55-s + ⋯ |
| L(s) = 1 | − 0.955·3-s − 0.0571·5-s − 0.0819·7-s − 0.0879·9-s + 0.315·11-s − 0.277·13-s + 0.0545·15-s − 1.44·17-s − 1.07·19-s + 0.0782·21-s + 1.47·23-s − 0.996·25-s + 1.03·27-s + 0.185·29-s − 0.117·31-s − 0.301·33-s + 0.00468·35-s − 0.367·37-s + 0.264·39-s − 0.794·41-s − 1.07·43-s + 0.00502·45-s + 1.35·47-s − 0.993·49-s + 1.38·51-s + 0.459·53-s − 0.0180·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7132567924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7132567924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 + 0.127T + 5T^{2} \) |
| 7 | \( 1 + 0.216T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 31 | \( 1 + 0.651T + 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 - 9.25T + 47T^{2} \) |
| 53 | \( 1 - 3.34T + 53T^{2} \) |
| 59 | \( 1 - 0.200T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 0.923T + 71T^{2} \) |
| 73 | \( 1 + 6.80T + 73T^{2} \) |
| 79 | \( 1 - 8.97T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 9.09T + 89T^{2} \) |
| 97 | \( 1 - 2.74T + 97T^{2} \) |
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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
−8.137019349574925159017691845968, −7.08282784021664298053496887136, −6.61807182314321024078736870454, −6.03824699853989423440379029824, −5.15476646819727494185517918607, −4.63412720294530218121901145618, −3.77229988810472001511236651102, −2.72600983947449690413161732093, −1.79084966498035865292309363666, −0.45259899134108496551831272403,
0.45259899134108496551831272403, 1.79084966498035865292309363666, 2.72600983947449690413161732093, 3.77229988810472001511236651102, 4.63412720294530218121901145618, 5.15476646819727494185517918607, 6.03824699853989423440379029824, 6.61807182314321024078736870454, 7.08282784021664298053496887136, 8.137019349574925159017691845968
