| 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 − 4·13-s + 14-s − 15-s + 16-s + 17-s − 18-s − 4·19-s − 20-s − 21-s + 2·22-s + 23-s − 24-s + 25-s + 4·26-s + 27-s − 28-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 − 1.10·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-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 + 0.784·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.73662688019337, −14.03112344402860, −13.42820260668080, −12.86921767799434, −12.50216768525979, −11.99693718085330, −11.49069498749048, −10.77246670865125, −10.37374190353781, −9.998388645427815, −9.359095179278379, −8.966058182120747, −8.399183971531302, −7.781807747701441, −7.647576348448239, −6.823785039332778, −6.549064845176256, −5.776509903600139, −4.916967684710496, −4.672994562754286, −3.742656126088552, −3.126693812592978, −2.745012539146785, −1.949608857622513, −1.382177897246375, 0, 0,
1.382177897246375, 1.949608857622513, 2.745012539146785, 3.126693812592978, 3.742656126088552, 4.672994562754286, 4.916967684710496, 5.776509903600139, 6.549064845176256, 6.823785039332778, 7.647576348448239, 7.781807747701441, 8.399183971531302, 8.966058182120747, 9.359095179278379, 9.998388645427815, 10.37374190353781, 10.77246670865125, 11.49069498749048, 11.99693718085330, 12.50216768525979, 12.86921767799434, 13.42820260668080, 14.03112344402860, 14.73662688019337
