| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 2·11-s − 12-s − 14-s + 15-s + 16-s − 5·17-s − 18-s − 2·19-s − 20-s − 21-s − 2·22-s + 23-s + 24-s + 25-s − 27-s + 28-s + 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.218·21-s − 0.426·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.250212693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250212693\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.18791423869031, −14.47816763142351, −13.97288024534738, −13.26429155103407, −12.67417273832008, −12.14495929405860, −11.69149358938801, −11.09497284466769, −10.83954360827336, −10.24214954336599, −9.560540183353599, −8.883919326029099, −8.688916293763955, −7.793278992519875, −7.466781619707261, −6.652698145461042, −6.367076557785265, −5.689201081369056, −4.803050751494267, −4.344204373946054, −3.756106659539579, −2.695138270869740, −2.153696374698099, −1.153166330411543, −0.5571645425380240,
0.5571645425380240, 1.153166330411543, 2.153696374698099, 2.695138270869740, 3.756106659539579, 4.344204373946054, 4.803050751494267, 5.689201081369056, 6.367076557785265, 6.652698145461042, 7.466781619707261, 7.793278992519875, 8.688916293763955, 8.883919326029099, 9.560540183353599, 10.24214954336599, 10.83954360827336, 11.09497284466769, 11.69149358938801, 12.14495929405860, 12.67417273832008, 13.26429155103407, 13.97288024534738, 14.47816763142351, 15.18791423869031
