| L(s) = 1 | − 2.70·2-s + 3·3-s − 0.701·4-s + 11.0·5-s − 8.10·6-s − 7·7-s + 23.5·8-s + 9·9-s − 29.8·10-s + 5.06·11-s − 2.10·12-s − 81.9·13-s + 18.9·14-s + 33.1·15-s − 57.8·16-s − 8.44·17-s − 24.3·18-s − 25.2·19-s − 7.75·20-s − 21·21-s − 13.6·22-s − 23·23-s + 70.5·24-s − 2.92·25-s + 221.·26-s + 27·27-s + 4.91·28-s + ⋯ |
| L(s) = 1 | − 0.955·2-s + 0.577·3-s − 0.0876·4-s + 0.988·5-s − 0.551·6-s − 0.377·7-s + 1.03·8-s + 0.333·9-s − 0.943·10-s + 0.138·11-s − 0.0506·12-s − 1.74·13-s + 0.361·14-s + 0.570·15-s − 0.904·16-s − 0.120·17-s − 0.318·18-s − 0.304·19-s − 0.0866·20-s − 0.218·21-s − 0.132·22-s − 0.208·23-s + 0.599·24-s − 0.0233·25-s + 1.66·26-s + 0.192·27-s + 0.0331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 2.70T + 8T^{2} \) |
| 5 | \( 1 - 11.0T + 125T^{2} \) |
| 11 | \( 1 - 5.06T + 1.33e3T^{2} \) |
| 13 | \( 1 + 81.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 8.44T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.2T + 6.85e3T^{2} \) |
| 29 | \( 1 - 67.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 25.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 54.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 39.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 45.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 78.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 689.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 971.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.07e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 282.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 129.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 678.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 539.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 663.T + 9.12e5T^{2} \) |
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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
−9.980131923520321640155096666115, −9.317708456719029474748460323971, −8.556085891082106349202328939646, −7.54647051998256044307912707361, −6.72489474754728443614729932487, −5.36323110422934097881086760294, −4.30157793903978839311626841540, −2.68143401051434301243510148679, −1.65343048168390364757529718047, 0,
1.65343048168390364757529718047, 2.68143401051434301243510148679, 4.30157793903978839311626841540, 5.36323110422934097881086760294, 6.72489474754728443614729932487, 7.54647051998256044307912707361, 8.556085891082106349202328939646, 9.317708456719029474748460323971, 9.980131923520321640155096666115
