| L(s) = 1 | − 2-s − 0.633·3-s + 4-s + 3.82·5-s + 0.633·6-s + 0.499·7-s − 8-s − 2.59·9-s − 3.82·10-s − 3.01·11-s − 0.633·12-s + 2.16·13-s − 0.499·14-s − 2.42·15-s + 16-s − 1.00·17-s + 2.59·18-s + 3.47·19-s + 3.82·20-s − 0.316·21-s + 3.01·22-s − 23-s + 0.633·24-s + 9.60·25-s − 2.16·26-s + 3.54·27-s + 0.499·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.366·3-s + 0.5·4-s + 1.70·5-s + 0.258·6-s + 0.188·7-s − 0.353·8-s − 0.866·9-s − 1.20·10-s − 0.907·11-s − 0.183·12-s + 0.600·13-s − 0.133·14-s − 0.625·15-s + 0.250·16-s − 0.243·17-s + 0.612·18-s + 0.796·19-s + 0.854·20-s − 0.0691·21-s + 0.641·22-s − 0.208·23-s + 0.129·24-s + 1.92·25-s − 0.424·26-s + 0.683·27-s + 0.0944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.611006269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611006269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.633T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 - 0.499T + 7T^{2} \) |
| 11 | \( 1 + 3.01T + 11T^{2} \) |
| 13 | \( 1 - 2.16T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 + 7.85T + 41T^{2} \) |
| 43 | \( 1 - 0.872T + 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 2.15T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 5.96T + 83T^{2} \) |
| 89 | \( 1 - 7.03T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.252886641819839687429493722916, −7.43392980423586461955121619840, −6.38950038421764714233496857883, −6.12011886099439389611100846778, −5.36043175598010508607547827293, −4.83848391223135035352421577788, −3.27254569583027450963695035811, −2.55741325518467604101039788359, −1.79815736269158166052038349287, −0.75882266541962608638719555635,
0.75882266541962608638719555635, 1.79815736269158166052038349287, 2.55741325518467604101039788359, 3.27254569583027450963695035811, 4.83848391223135035352421577788, 5.36043175598010508607547827293, 6.12011886099439389611100846778, 6.38950038421764714233496857883, 7.43392980423586461955121619840, 8.252886641819839687429493722916
