| L(s) = 1 | + 1.09·3-s − 5-s − 2.80·7-s − 1.80·9-s + 2.45·11-s + 3.55·13-s − 1.09·15-s − 4.34·17-s − 19-s − 3.06·21-s + 1.19·23-s + 25-s − 5.25·27-s + 7.44·29-s + 2.91·31-s + 2.68·33-s + 2.80·35-s + 7.89·37-s + 3.88·39-s − 10.5·41-s + 2.95·43-s + 1.80·45-s + 13.1·47-s + 0.845·49-s − 4.75·51-s + 3.74·53-s − 2.45·55-s + ⋯ |
| L(s) = 1 | + 0.632·3-s − 0.447·5-s − 1.05·7-s − 0.600·9-s + 0.740·11-s + 0.984·13-s − 0.282·15-s − 1.05·17-s − 0.229·19-s − 0.669·21-s + 0.250·23-s + 0.200·25-s − 1.01·27-s + 1.38·29-s + 0.523·31-s + 0.468·33-s + 0.473·35-s + 1.29·37-s + 0.622·39-s − 1.64·41-s + 0.450·43-s + 0.268·45-s + 1.91·47-s + 0.120·49-s − 0.666·51-s + 0.513·53-s − 0.331·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719270089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719270089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.09T + 3T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 - 7.44T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 2.95T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 2.45T + 61T^{2} \) |
| 67 | \( 1 + 9.59T + 67T^{2} \) |
| 71 | \( 1 + 7.60T + 71T^{2} \) |
| 73 | \( 1 - 2.75T + 73T^{2} \) |
| 79 | \( 1 - 9.79T + 79T^{2} \) |
| 83 | \( 1 - 6.95T + 83T^{2} \) |
| 89 | \( 1 - 8.49T + 89T^{2} \) |
| 97 | \( 1 - 2.01T + 97T^{2} \) |
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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
−8.778857774679386445825970822939, −8.197415859258520608683184467275, −7.16501924952996723643791733072, −6.43008187213180767383469118791, −5.94969110076985092628436029925, −4.62326830523207137709662596325, −3.81818948258221125358358272104, −3.15245234625981408337246521820, −2.29912738106816343959111198444, −0.76040197716327222908052154906,
0.76040197716327222908052154906, 2.29912738106816343959111198444, 3.15245234625981408337246521820, 3.81818948258221125358358272104, 4.62326830523207137709662596325, 5.94969110076985092628436029925, 6.43008187213180767383469118791, 7.16501924952996723643791733072, 8.197415859258520608683184467275, 8.778857774679386445825970822939
