| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s + 3·7-s − 3·8-s − 2·9-s − 10-s − 12-s − 5·13-s + 3·14-s − 15-s − 16-s + 3·17-s − 2·18-s + 19-s + 20-s + 3·21-s − 7·23-s − 3·24-s − 4·25-s − 5·26-s − 5·27-s − 3·28-s + 3·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.13·7-s − 1.06·8-s − 2/3·9-s − 0.316·10-s − 0.288·12-s − 1.38·13-s + 0.801·14-s − 0.258·15-s − 1/4·16-s + 0.727·17-s − 0.471·18-s + 0.229·19-s + 0.223·20-s + 0.654·21-s − 1.45·23-s − 0.612·24-s − 4/5·25-s − 0.980·26-s − 0.962·27-s − 0.566·28-s + 0.557·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.584979645590685413731256190470, −8.147322815006664698727426415156, −7.54828976874341140852447073222, −6.27874266213962806152052320920, −5.25796596193718363670664066003, −4.82527890316393394245156789173, −3.83743562231691862277531207594, −3.04471622817420247386835239693, −1.93738046601125160840719041634, 0,
1.93738046601125160840719041634, 3.04471622817420247386835239693, 3.83743562231691862277531207594, 4.82527890316393394245156789173, 5.25796596193718363670664066003, 6.27874266213962806152052320920, 7.54828976874341140852447073222, 8.147322815006664698727426415156, 8.584979645590685413731256190470
