| L(s) = 1 | + 2.75·2-s + 5.59·4-s + 5-s + 0.393·7-s + 9.92·8-s + 2.75·10-s + 0.393·11-s − 2.56·13-s + 1.08·14-s + 16.1·16-s − 2.07·17-s − 0.958·19-s + 5.59·20-s + 1.08·22-s − 6.15·23-s + 25-s − 7.07·26-s + 2.20·28-s + 29-s − 10.1·31-s + 24.6·32-s − 5.72·34-s + 0.393·35-s − 7.34·37-s − 2.64·38-s + 9.92·40-s + 1.65·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.79·4-s + 0.447·5-s + 0.148·7-s + 3.50·8-s + 0.871·10-s + 0.118·11-s − 0.711·13-s + 0.290·14-s + 4.03·16-s − 0.504·17-s − 0.219·19-s + 1.25·20-s + 0.231·22-s − 1.28·23-s + 0.200·25-s − 1.38·26-s + 0.416·28-s + 0.185·29-s − 1.82·31-s + 4.36·32-s − 0.982·34-s + 0.0665·35-s − 1.20·37-s − 0.428·38-s + 1.56·40-s + 0.258·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.037967476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.037967476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 7 | \( 1 - 0.393T + 7T^{2} \) |
| 11 | \( 1 - 0.393T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + 0.958T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 9.54T + 59T^{2} \) |
| 61 | \( 1 + 6.25T + 61T^{2} \) |
| 67 | \( 1 + 7.42T + 67T^{2} \) |
| 71 | \( 1 + 5.98T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 6.41T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−9.998729396058834736373674510196, −8.773854017889706828390774493227, −7.48763439957280655097502882799, −7.00019248643794014659916300158, −5.93146675114601501052917610373, −5.48787434677885647395306902480, −4.46522086381216913223976409713, −3.79887005819955356746062876026, −2.59164670677942975261980067788, −1.84436193522081983643835915560,
1.84436193522081983643835915560, 2.59164670677942975261980067788, 3.79887005819955356746062876026, 4.46522086381216913223976409713, 5.48787434677885647395306902480, 5.93146675114601501052917610373, 7.00019248643794014659916300158, 7.48763439957280655097502882799, 8.773854017889706828390774493227, 9.998729396058834736373674510196
