| L(s) = 1 | + 8.94·3-s + 5·5-s − 32.7·7-s + 52.9·9-s − 45.7·11-s + 45.5·13-s + 44.7·15-s − 76.5·17-s + 32.0·19-s − 293.·21-s − 23·23-s + 25·25-s + 232.·27-s + 287.·29-s + 161.·31-s − 409.·33-s − 163.·35-s + 7.53·37-s + 407.·39-s + 216.·41-s + 469.·43-s + 264.·45-s + 210.·47-s + 730.·49-s − 684.·51-s + 58.7·53-s − 228.·55-s + ⋯ |
| L(s) = 1 | + 1.72·3-s + 0.447·5-s − 1.76·7-s + 1.96·9-s − 1.25·11-s + 0.972·13-s + 0.769·15-s − 1.09·17-s + 0.386·19-s − 3.04·21-s − 0.208·23-s + 0.200·25-s + 1.65·27-s + 1.83·29-s + 0.936·31-s − 2.15·33-s − 0.791·35-s + 0.0334·37-s + 1.67·39-s + 0.823·41-s + 1.66·43-s + 0.877·45-s + 0.652·47-s + 2.12·49-s − 1.87·51-s + 0.152·53-s − 0.560·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.727753151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.727753151\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 8.94T + 27T^{2} \) |
| 7 | \( 1 + 32.7T + 343T^{2} \) |
| 11 | \( 1 + 45.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.0T + 6.85e3T^{2} \) |
| 29 | \( 1 - 287.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 7.53T + 5.06e4T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 469.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 210.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 58.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 376.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 52.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 16.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 209.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 811.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 63.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 436.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 655.T + 9.12e5T^{2} \) |
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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
−8.921963197654178093562866319851, −8.296085937166820032294721146060, −7.42631092537568134781926636153, −6.59355140326850760580790900463, −5.90584180732628663277668936411, −4.50561802114048705702844294047, −3.60795758996711955335859338440, −2.74801252452625671302720340078, −2.42528560481600675660778479463, −0.812710785212460424056256829853,
0.812710785212460424056256829853, 2.42528560481600675660778479463, 2.74801252452625671302720340078, 3.60795758996711955335859338440, 4.50561802114048705702844294047, 5.90584180732628663277668936411, 6.59355140326850760580790900463, 7.42631092537568134781926636153, 8.296085937166820032294721146060, 8.921963197654178093562866319851
