| L(s) = 1 | − 1.51·3-s + 0.798·5-s − 0.693·9-s − 5.64·11-s − 4.80·13-s − 1.21·15-s + 17-s + 5.07·19-s + 4.19·23-s − 4.36·25-s + 5.60·27-s + 1.85·29-s − 10.0·31-s + 8.57·33-s − 6.11·37-s + 7.28·39-s − 6.24·41-s − 0.575·43-s − 0.553·45-s − 5.74·47-s − 1.51·51-s + 4.50·53-s − 4.50·55-s − 7.71·57-s − 7.97·59-s − 8.93·61-s − 3.83·65-s + ⋯ |
| L(s) = 1 | − 0.876·3-s + 0.357·5-s − 0.231·9-s − 1.70·11-s − 1.33·13-s − 0.313·15-s + 0.242·17-s + 1.16·19-s + 0.873·23-s − 0.872·25-s + 1.07·27-s + 0.345·29-s − 1.81·31-s + 1.49·33-s − 1.00·37-s + 1.16·39-s − 0.976·41-s − 0.0878·43-s − 0.0825·45-s − 0.837·47-s − 0.212·51-s + 0.618·53-s − 0.607·55-s − 1.02·57-s − 1.03·59-s − 1.14·61-s − 0.475·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6104808173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6104808173\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.51T + 3T^{2} \) |
| 5 | \( 1 - 0.798T + 5T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 6.11T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 0.575T + 43T^{2} \) |
| 47 | \( 1 + 5.74T + 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 + 8.93T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 + 5.24T + 83T^{2} \) |
| 89 | \( 1 - 0.528T + 89T^{2} \) |
| 97 | \( 1 + 5.53T + 97T^{2} \) |
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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
−7.78306540483682981157787319508, −7.33313907025068002798549871633, −6.56736024138772401878226536773, −5.56014506910103126203240073528, −5.24715207411849469716525166071, −4.87011209395745779546492630117, −3.44051186112389900536376740720, −2.74459654278748793104497587073, −1.83824298721347343724993657736, −0.39934263734339154603177015109,
0.39934263734339154603177015109, 1.83824298721347343724993657736, 2.74459654278748793104497587073, 3.44051186112389900536376740720, 4.87011209395745779546492630117, 5.24715207411849469716525166071, 5.56014506910103126203240073528, 6.56736024138772401878226536773, 7.33313907025068002798549871633, 7.78306540483682981157787319508
