| L(s) = 1 | − 3.46·2-s − 7·3-s + 3.99·4-s − 13.8·5-s + 24.2·6-s + 22.5·7-s + 13.8·8-s + 22·9-s + 47.9·10-s + 22.5·11-s − 27.9·12-s − 77.9·14-s + 96.9·15-s − 80·16-s − 27·17-s − 76.2·18-s + 88.3·19-s − 55.4·20-s − 157.·21-s − 77.9·22-s − 57·23-s − 96.9·24-s + 66.9·25-s + 35·27-s + 90.0·28-s − 69·29-s − 336·30-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 1.34·3-s + 0.499·4-s − 1.23·5-s + 1.64·6-s + 1.21·7-s + 0.612·8-s + 0.814·9-s + 1.51·10-s + 0.617·11-s − 0.673·12-s − 1.48·14-s + 1.66·15-s − 1.25·16-s − 0.385·17-s − 0.997·18-s + 1.06·19-s − 0.619·20-s − 1.63·21-s − 0.755·22-s − 0.516·23-s − 0.824·24-s + 0.535·25-s + 0.249·27-s + 0.607·28-s − 0.441·29-s − 2.04·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 + 3.46T + 8T^{2} \) |
| 3 | \( 1 + 7T + 27T^{2} \) |
| 5 | \( 1 + 13.8T + 125T^{2} \) |
| 7 | \( 1 - 22.5T + 343T^{2} \) |
| 11 | \( 1 - 22.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 27T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 57T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69T + 2.43e4T^{2} \) |
| 31 | \( 1 - 72.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 39.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 85T + 7.95e4T^{2} \) |
| 47 | \( 1 + 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 426T + 1.48e5T^{2} \) |
| 59 | \( 1 + 19.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 17T + 2.26e5T^{2} \) |
| 67 | \( 1 - 164.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 306.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−11.48217719835871621332339167537, −11.06294483507506560520482804712, −9.921891229696085597481046122789, −8.581143343426175824590953335122, −7.80199539476799250738132440759, −6.83753034594576023534015011970, −5.20560348401395186663010199590, −4.19252532384516695831971535487, −1.29137390609072600850336473452, 0,
1.29137390609072600850336473452, 4.19252532384516695831971535487, 5.20560348401395186663010199590, 6.83753034594576023534015011970, 7.80199539476799250738132440759, 8.581143343426175824590953335122, 9.921891229696085597481046122789, 11.06294483507506560520482804712, 11.48217719835871621332339167537
