| L(s) = 1 | − 0.381·2-s − 1.85·4-s − 5-s + 7-s + 1.47·8-s + 0.381·10-s + 11-s + 0.236·13-s − 0.381·14-s + 3.14·16-s − 6.47·17-s + 3.47·19-s + 1.85·20-s − 0.381·22-s − 8.47·23-s − 4·25-s − 0.0901·26-s − 1.85·28-s − 1.76·29-s − 0.472·31-s − 4.14·32-s + 2.47·34-s − 35-s − 5.47·37-s − 1.32·38-s − 1.47·40-s − 4.47·41-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.927·4-s − 0.447·5-s + 0.377·7-s + 0.520·8-s + 0.120·10-s + 0.301·11-s + 0.0654·13-s − 0.102·14-s + 0.786·16-s − 1.56·17-s + 0.796·19-s + 0.414·20-s − 0.0814·22-s − 1.76·23-s − 0.800·25-s − 0.0176·26-s − 0.350·28-s − 0.327·29-s − 0.0847·31-s − 0.732·32-s + 0.423·34-s − 0.169·35-s − 0.899·37-s − 0.215·38-s − 0.232·40-s − 0.698·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 - 0.236T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 + 8.47T + 23T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 0.236T + 47T^{2} \) |
| 53 | \( 1 + 5.52T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 + 9.76T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.888912523750348806547038670875, −9.157918330307581212586057256194, −8.304824023050948723963500609033, −7.69796390449055974908492960853, −6.53100160719415771482904328931, −5.35447979095120156599135967404, −4.39158162458797089406994311472, −3.63104676949937777392394230778, −1.82200208131314142461308596676, 0,
1.82200208131314142461308596676, 3.63104676949937777392394230778, 4.39158162458797089406994311472, 5.35447979095120156599135967404, 6.53100160719415771482904328931, 7.69796390449055974908492960853, 8.304824023050948723963500609033, 9.157918330307581212586057256194, 9.888912523750348806547038670875
