| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 11-s + 13-s + 14-s + 16-s + 6·17-s − 4·19-s + 20-s − 22-s + 25-s + 26-s + 28-s + 6·29-s + 8·31-s + 32-s + 6·34-s + 35-s − 10·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s − 0.213·22-s + 1/5·25-s + 0.196·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s − 1.64·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.388811864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.388811864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.96150037545326, −13.51618896154174, −12.84330547287609, −12.33451621580937, −12.12727881028180, −11.51774788356765, −10.74419993024673, −10.58540616986603, −10.00836800630883, −9.499809457886080, −8.751119490275260, −8.254818010563151, −7.830871941761953, −7.237689896240559, −6.495419938276744, −6.191559018173428, −5.638686020107332, −4.893645419427958, −4.730982194051607, −3.922509726470623, −3.213090255871888, −2.835310957037980, −2.001669868456192, −1.450641330031811, −0.6610190136397434,
0.6610190136397434, 1.450641330031811, 2.001669868456192, 2.835310957037980, 3.213090255871888, 3.922509726470623, 4.730982194051607, 4.893645419427958, 5.638686020107332, 6.191559018173428, 6.495419938276744, 7.237689896240559, 7.830871941761953, 8.254818010563151, 8.751119490275260, 9.499809457886080, 10.00836800630883, 10.58540616986603, 10.74419993024673, 11.51774788356765, 12.12727881028180, 12.33451621580937, 12.84330547287609, 13.51618896154174, 13.96150037545326
