| L(s) = 1 | + 2-s + 4-s + 5-s + 5.07·7-s + 8-s + 10-s − 0.530·11-s + 6.47·13-s + 5.07·14-s + 16-s − 6.46·17-s + 3.29·19-s + 20-s − 0.530·22-s − 7.65·23-s + 25-s + 6.47·26-s + 5.07·28-s + 7.33·29-s + 5.67·31-s + 32-s − 6.46·34-s + 5.07·35-s − 4.14·37-s + 3.29·38-s + 40-s − 3.34·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.91·7-s + 0.353·8-s + 0.316·10-s − 0.159·11-s + 1.79·13-s + 1.35·14-s + 0.250·16-s − 1.56·17-s + 0.755·19-s + 0.223·20-s − 0.113·22-s − 1.59·23-s + 0.200·25-s + 1.26·26-s + 0.958·28-s + 1.36·29-s + 1.01·31-s + 0.176·32-s − 1.10·34-s + 0.857·35-s − 0.682·37-s + 0.533·38-s + 0.158·40-s − 0.523·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.191193539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.191193539\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 7 | \( 1 - 5.07T + 7T^{2} \) |
| 11 | \( 1 + 0.530T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 7.33T + 29T^{2} \) |
| 31 | \( 1 - 5.67T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 + 9.16T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 3.11T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 0.638T + 67T^{2} \) |
| 71 | \( 1 - 6.84T + 71T^{2} \) |
| 73 | \( 1 + 6.26T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 + 0.366T + 83T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.036250050070249724927766752653, −6.91310532940655610067294640189, −6.41932797296055712459856307489, −5.58732372977578835212989604652, −5.05067307222314931603339551691, −4.32448219746954405812841319236, −3.76363240528689988053368656077, −2.55745522874089903424839702634, −1.83435980695348845395144148020, −1.12691303066033277045257198120,
1.12691303066033277045257198120, 1.83435980695348845395144148020, 2.55745522874089903424839702634, 3.76363240528689988053368656077, 4.32448219746954405812841319236, 5.05067307222314931603339551691, 5.58732372977578835212989604652, 6.41932797296055712459856307489, 6.91310532940655610067294640189, 8.036250050070249724927766752653
