| L(s) = 1 | − 3.39·3-s + 0.805·5-s − 1.94·7-s + 8.49·9-s − 0.661·11-s − 5.64·13-s − 2.73·15-s + 2·17-s + 2.15·19-s + 6.60·21-s − 3.59·23-s − 4.35·25-s − 18.6·27-s − 5.64·29-s + 6.32·31-s + 2.24·33-s − 1.56·35-s + 37-s + 19.1·39-s + 4.95·41-s + 5.04·43-s + 6.84·45-s + 3.51·47-s − 3.21·49-s − 6.78·51-s + 6.39·53-s − 0.532·55-s + ⋯ |
| L(s) = 1 | − 1.95·3-s + 0.360·5-s − 0.735·7-s + 2.83·9-s − 0.199·11-s − 1.56·13-s − 0.704·15-s + 0.485·17-s + 0.493·19-s + 1.44·21-s − 0.749·23-s − 0.870·25-s − 3.58·27-s − 1.04·29-s + 1.13·31-s + 0.390·33-s − 0.264·35-s + 0.164·37-s + 3.06·39-s + 0.773·41-s + 0.768·43-s + 1.02·45-s + 0.512·47-s − 0.458·49-s − 0.949·51-s + 0.878·53-s − 0.0717·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5915976114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5915976114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 3.39T + 3T^{2} \) |
| 5 | \( 1 - 0.805T + 5T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 + 0.661T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 + 5.64T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 41 | \( 1 - 4.95T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 - 6.39T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 3.59T + 79T^{2} \) |
| 83 | \( 1 - 4.83T + 83T^{2} \) |
| 89 | \( 1 - 5.51T + 89T^{2} \) |
| 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
−9.892362224248710824308465579481, −9.507460432277842887079740922327, −7.71539560381428929536769911814, −7.13413127659600572072033874716, −6.20208334371818918202344312218, −5.62969489610139524966436442336, −4.89608366293133176570030937882, −3.89092634801377686919165730736, −2.21866623721386306547064900798, −0.61698057409065557564211508756,
0.61698057409065557564211508756, 2.21866623721386306547064900798, 3.89092634801377686919165730736, 4.89608366293133176570030937882, 5.62969489610139524966436442336, 6.20208334371818918202344312218, 7.13413127659600572072033874716, 7.71539560381428929536769911814, 9.507460432277842887079740922327, 9.892362224248710824308465579481
