| L(s) = 1 | − 1.69·2-s − 3-s + 0.868·4-s + 3.51·5-s + 1.69·6-s − 7-s + 1.91·8-s + 9-s − 5.94·10-s − 1.74·11-s − 0.868·12-s + 6.33·13-s + 1.69·14-s − 3.51·15-s − 4.98·16-s − 5.94·17-s − 1.69·18-s + 1.74·19-s + 3.04·20-s + 21-s + 2.95·22-s + 23-s − 1.91·24-s + 7.33·25-s − 10.7·26-s − 27-s − 0.868·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 0.577·3-s + 0.434·4-s + 1.57·5-s + 0.691·6-s − 0.377·7-s + 0.677·8-s + 0.333·9-s − 1.88·10-s − 0.525·11-s − 0.250·12-s + 1.75·13-s + 0.452·14-s − 0.906·15-s − 1.24·16-s − 1.44·17-s − 0.399·18-s + 0.399·19-s + 0.681·20-s + 0.218·21-s + 0.629·22-s + 0.208·23-s − 0.391·24-s + 1.46·25-s − 2.10·26-s − 0.192·27-s − 0.164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7919422528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7919422528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 - 6.33T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 7.94T + 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 8.69T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 + 7.42T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 - 4.18T + 89T^{2} \) |
| 97 | \( 1 - 5.53T + 97T^{2} \) |
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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
−10.71467008332142268420101746969, −10.06522826336291864322771359771, −9.183795088291003891715773775501, −8.697698004795462907973006810710, −7.37047365136649322702300170112, −6.30731291380189075104245450433, −5.70672545216908292895950509535, −4.33552640817428854407863321065, −2.36780115030343750314389151899, −1.07446734187228023635943245477,
1.07446734187228023635943245477, 2.36780115030343750314389151899, 4.33552640817428854407863321065, 5.70672545216908292895950509535, 6.30731291380189075104245450433, 7.37047365136649322702300170112, 8.697698004795462907973006810710, 9.183795088291003891715773775501, 10.06522826336291864322771359771, 10.71467008332142268420101746969
