| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 1.69·7-s − 8-s + 9-s + 10-s − 2.15·11-s − 12-s − 1.69·14-s + 15-s + 16-s + 2.35·17-s − 18-s − 0.198·19-s − 20-s − 1.69·21-s + 2.15·22-s − 3.74·23-s + 24-s + 25-s − 27-s + 1.69·28-s + 1.29·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.639·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.650·11-s − 0.288·12-s − 0.452·14-s + 0.258·15-s + 0.250·16-s + 0.571·17-s − 0.235·18-s − 0.0454·19-s − 0.223·20-s − 0.369·21-s + 0.460·22-s − 0.780·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.319·28-s + 0.240·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 0.198T + 19T^{2} \) |
| 23 | \( 1 + 3.74T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 + 0.801T + 37T^{2} \) |
| 41 | \( 1 - 1.89T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 + 1.00T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 + 6.32T + 61T^{2} \) |
| 67 | \( 1 - 7.56T + 67T^{2} \) |
| 71 | \( 1 - 4.18T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 8.43T + 83T^{2} \) |
| 89 | \( 1 - 2.41T + 89T^{2} \) |
| 97 | \( 1 - 0.0881T + 97T^{2} \) |
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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
−7.81595113425258159992284898349, −7.43394452507102904464981692396, −6.47304666325712688342873988536, −5.76777873135145552171087500729, −5.00848434070333917900898865719, −4.22853220196205944415798801493, −3.21734791240840005756304177954, −2.17147228977193801370508625623, −1.16038507217830279041073955164, 0,
1.16038507217830279041073955164, 2.17147228977193801370508625623, 3.21734791240840005756304177954, 4.22853220196205944415798801493, 5.00848434070333917900898865719, 5.76777873135145552171087500729, 6.47304666325712688342873988536, 7.43394452507102904464981692396, 7.81595113425258159992284898349
