| L(s) = 1 | + 4.58·2-s − 3·3-s + 12.9·4-s − 17.0·5-s − 13.7·6-s − 20.9·7-s + 22.8·8-s + 9·9-s − 77.9·10-s − 48.1·11-s − 38.9·12-s + 29.4·13-s − 96.1·14-s + 51.0·15-s + 0.636·16-s + 108.·17-s + 41.2·18-s − 111.·19-s − 220.·20-s + 62.9·21-s − 220.·22-s − 14.3·23-s − 68.4·24-s + 164.·25-s + 134.·26-s − 27·27-s − 272.·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 0.577·3-s + 1.62·4-s − 1.52·5-s − 0.934·6-s − 1.13·7-s + 1.00·8-s + 0.333·9-s − 2.46·10-s − 1.32·11-s − 0.936·12-s + 0.628·13-s − 1.83·14-s + 0.878·15-s + 0.00994·16-s + 1.54·17-s + 0.539·18-s − 1.34·19-s − 2.46·20-s + 0.654·21-s − 2.13·22-s − 0.130·23-s − 0.581·24-s + 1.31·25-s + 1.01·26-s − 0.192·27-s − 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 4.58T + 8T^{2} \) |
| 5 | \( 1 + 17.0T + 125T^{2} \) |
| 7 | \( 1 + 20.9T + 343T^{2} \) |
| 11 | \( 1 + 48.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 14.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 292.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 216.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 342.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 325.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 315.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 256.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 749.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 664.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 207.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 45.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 813.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 19.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 636.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 235.T + 9.12e5T^{2} \) |
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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
−12.24322666218727003694005816079, −11.09597708495705090801282400495, −10.27073586294479759064694156381, −8.324040983380215526919243669838, −7.15745668259941798909254992798, −6.14939661296469118237415473262, −5.05287762622093793736479334131, −3.88879478007538630779664720436, −3.05841182362752964719482326657, 0,
3.05841182362752964719482326657, 3.88879478007538630779664720436, 5.05287762622093793736479334131, 6.14939661296469118237415473262, 7.15745668259941798909254992798, 8.324040983380215526919243669838, 10.27073586294479759064694156381, 11.09597708495705090801282400495, 12.24322666218727003694005816079
