| L(s) = 1 | − 0.517·2-s + 3-s − 1.73·4-s − 0.162·5-s − 0.517·6-s − 5.01·7-s + 1.93·8-s + 9-s + 0.0839·10-s + 0.214·11-s − 1.73·12-s + 4.41·13-s + 2.59·14-s − 0.162·15-s + 2.46·16-s − 17-s − 0.517·18-s − 3.47·19-s + 0.281·20-s − 5.01·21-s − 0.111·22-s − 3.31·23-s + 1.93·24-s − 4.97·25-s − 2.28·26-s + 27-s + 8.68·28-s + ⋯ |
| L(s) = 1 | − 0.365·2-s + 0.577·3-s − 0.866·4-s − 0.0725·5-s − 0.211·6-s − 1.89·7-s + 0.682·8-s + 0.333·9-s + 0.0265·10-s + 0.0647·11-s − 0.500·12-s + 1.22·13-s + 0.693·14-s − 0.0419·15-s + 0.616·16-s − 0.242·17-s − 0.121·18-s − 0.797·19-s + 0.0628·20-s − 1.09·21-s − 0.0236·22-s − 0.691·23-s + 0.394·24-s − 0.994·25-s − 0.447·26-s + 0.192·27-s + 1.64·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8641227300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8641227300\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.517T + 2T^{2} \) |
| 5 | \( 1 + 0.162T + 5T^{2} \) |
| 7 | \( 1 + 5.01T + 7T^{2} \) |
| 11 | \( 1 - 0.214T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 - 0.557T + 31T^{2} \) |
| 37 | \( 1 + 9.27T + 37T^{2} \) |
| 41 | \( 1 + 8.10T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 7.37T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 3.95T + 73T^{2} \) |
| 83 | \( 1 - 8.13T + 83T^{2} \) |
| 89 | \( 1 - 2.43T + 89T^{2} \) |
| 97 | \( 1 - 0.816T + 97T^{2} \) |
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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
−8.713320812926620448783753725757, −7.88852230669730000177812604600, −7.02922032003776349433048481672, −6.25661294858591732689427997319, −5.67123055947925432198333815245, −4.33821086313314378511504972137, −3.77146346543192524749847015980, −3.17714223539175656474516838694, −1.93697796312978422117642891154, −0.53756149297479227602787324889,
0.53756149297479227602787324889, 1.93697796312978422117642891154, 3.17714223539175656474516838694, 3.77146346543192524749847015980, 4.33821086313314378511504972137, 5.67123055947925432198333815245, 6.25661294858591732689427997319, 7.02922032003776349433048481672, 7.88852230669730000177812604600, 8.713320812926620448783753725757
