| L(s) = 1 | − 3.76·2-s + 6.16·4-s + 15.4·5-s − 7·7-s + 6.90·8-s − 58.3·10-s − 11·11-s + 49.0·13-s + 26.3·14-s − 75.3·16-s + 34.1·17-s − 144.·19-s + 95.5·20-s + 41.4·22-s − 118.·23-s + 115.·25-s − 184.·26-s − 43.1·28-s + 63.6·29-s − 212.·31-s + 228.·32-s − 128.·34-s − 108.·35-s − 200.·37-s + 542.·38-s + 106.·40-s − 451.·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.770·4-s + 1.38·5-s − 0.377·7-s + 0.305·8-s − 1.84·10-s − 0.301·11-s + 1.04·13-s + 0.502·14-s − 1.17·16-s + 0.486·17-s − 1.74·19-s + 1.06·20-s + 0.401·22-s − 1.07·23-s + 0.920·25-s − 1.39·26-s − 0.291·28-s + 0.407·29-s − 1.23·31-s + 1.26·32-s − 0.647·34-s − 0.523·35-s − 0.889·37-s + 2.31·38-s + 0.422·40-s − 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 3.76T + 8T^{2} \) |
| 5 | \( 1 - 15.4T + 125T^{2} \) |
| 13 | \( 1 - 49.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 34.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 63.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 200.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 130.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 176.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 629.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 86.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 400.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 507.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 176.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 701.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 788.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 185.T + 9.12e5T^{2} \) |
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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
−9.674147702552499097156771019587, −8.752444005263353151395260885725, −8.335266626450144470910780602392, −7.02836232436908219879105323332, −6.25607465360071010486670127776, −5.37709473805517550820976300062, −3.89631367850701075533333192742, −2.27790683984819339740675165412, −1.48526244251488619030461131691, 0,
1.48526244251488619030461131691, 2.27790683984819339740675165412, 3.89631367850701075533333192742, 5.37709473805517550820976300062, 6.25607465360071010486670127776, 7.02836232436908219879105323332, 8.335266626450144470910780602392, 8.752444005263353151395260885725, 9.674147702552499097156771019587
