| L(s) = 1 | − 0.719·2-s + 3-s − 1.48·4-s + 1.82·5-s − 0.719·6-s + 0.414·7-s + 2.50·8-s + 9-s − 1.31·10-s − 2.75·11-s − 1.48·12-s + 0.245·13-s − 0.298·14-s + 1.82·15-s + 1.15·16-s + 17-s − 0.719·18-s + 0.386·19-s − 2.71·20-s + 0.414·21-s + 1.98·22-s + 6.58·23-s + 2.50·24-s − 1.65·25-s − 0.176·26-s + 27-s − 0.614·28-s + ⋯ |
| L(s) = 1 | − 0.508·2-s + 0.577·3-s − 0.740·4-s + 0.818·5-s − 0.293·6-s + 0.156·7-s + 0.886·8-s + 0.333·9-s − 0.416·10-s − 0.832·11-s − 0.427·12-s + 0.0681·13-s − 0.0797·14-s + 0.472·15-s + 0.289·16-s + 0.242·17-s − 0.169·18-s + 0.0886·19-s − 0.606·20-s + 0.0905·21-s + 0.423·22-s + 1.37·23-s + 0.511·24-s − 0.330·25-s − 0.0346·26-s + 0.192·27-s − 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.767048325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.767048325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.719T + 2T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 - 0.414T + 7T^{2} \) |
| 11 | \( 1 + 2.75T + 11T^{2} \) |
| 13 | \( 1 - 0.245T + 13T^{2} \) |
| 19 | \( 1 - 0.386T + 19T^{2} \) |
| 23 | \( 1 - 6.58T + 23T^{2} \) |
| 29 | \( 1 + 0.775T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 - 7.95T + 37T^{2} \) |
| 41 | \( 1 + 4.65T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 59 | \( 1 - 2.36T + 59T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 - 9.15T + 67T^{2} \) |
| 71 | \( 1 + 3.23T + 71T^{2} \) |
| 73 | \( 1 + 3.86T + 73T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 0.214T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−8.476907287149823657082666403651, −7.917004666507085227025489292047, −7.23616666472074169492008437734, −6.25461558526168835587736226467, −5.29226110778074461496593027116, −4.82471655141435838099510779288, −3.78877765740468332612222239804, −2.83586911066810752901695186327, −1.89363001016625889877573993108, −0.829730491556810945611693655345,
0.829730491556810945611693655345, 1.89363001016625889877573993108, 2.83586911066810752901695186327, 3.78877765740468332612222239804, 4.82471655141435838099510779288, 5.29226110778074461496593027116, 6.25461558526168835587736226467, 7.23616666472074169492008437734, 7.917004666507085227025489292047, 8.476907287149823657082666403651
