| L(s) = 1 | − 3-s − 5-s − 4.32·7-s + 9-s + 1.39·11-s + 13-s + 15-s + 5.11·17-s − 2.92·19-s + 4.32·21-s − 8.97·23-s + 25-s − 27-s − 8.36·29-s − 4·31-s − 1.39·33-s + 4.32·35-s + 3.39·37-s − 39-s − 0.601·41-s + 2.79·43-s − 45-s + 9.29·47-s + 11.6·49-s − 5.11·51-s − 2.19·53-s − 1.39·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.63·7-s + 0.333·9-s + 0.421·11-s + 0.277·13-s + 0.258·15-s + 1.24·17-s − 0.671·19-s + 0.943·21-s − 1.87·23-s + 0.200·25-s − 0.192·27-s − 1.55·29-s − 0.718·31-s − 0.243·33-s + 0.730·35-s + 0.558·37-s − 0.160·39-s − 0.0939·41-s + 0.426·43-s − 0.149·45-s + 1.35·47-s + 1.67·49-s − 0.716·51-s − 0.301·53-s − 0.188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6459314370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6459314370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4.32T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 + 8.97T + 23T^{2} \) |
| 29 | \( 1 + 8.36T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 + 0.601T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 - 9.29T + 47T^{2} \) |
| 53 | \( 1 + 2.19T + 53T^{2} \) |
| 59 | \( 1 + 7.44T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 1.95T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.72231570435941109248173367069, −7.46761462191452391235455593732, −6.36576272413997351637909058476, −6.09679537805940822595706969829, −5.42048069451708364182279713237, −4.12110976583487756346814198119, −3.80100929852551537698359533036, −2.94608010752171278207937495414, −1.73997531847037311384829487592, −0.42265836307824378590430918545,
0.42265836307824378590430918545, 1.73997531847037311384829487592, 2.94608010752171278207937495414, 3.80100929852551537698359533036, 4.12110976583487756346814198119, 5.42048069451708364182279713237, 6.09679537805940822595706969829, 6.36576272413997351637909058476, 7.46761462191452391235455593732, 7.72231570435941109248173367069
