| L(s) = 1 | − 2·4-s − 3·9-s − 11-s + 3·13-s + 4·16-s + 5·17-s + 7·23-s − 5·25-s − 9·31-s + 6·36-s − 11·41-s + 2·44-s − 4·47-s − 7·49-s − 6·52-s − 13·53-s + 8·59-s − 8·64-s − 15·67-s − 10·68-s + 12·79-s + 9·81-s − 17·83-s − 14·92-s − 97-s + 3·99-s + 10·100-s + ⋯ |
| L(s) = 1 | − 4-s − 9-s − 0.301·11-s + 0.832·13-s + 16-s + 1.21·17-s + 1.45·23-s − 25-s − 1.61·31-s + 36-s − 1.71·41-s + 0.301·44-s − 0.583·47-s − 49-s − 0.832·52-s − 1.78·53-s + 1.04·59-s − 64-s − 1.83·67-s − 1.21·68-s + 1.35·79-s + 81-s − 1.86·83-s − 1.45·92-s − 0.101·97-s + 0.301·99-s + 100-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 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.836974720135206255610544472959, −8.203981792922301940504110098742, −7.49774877417135048713191737937, −6.27013422173559366918615833585, −5.47324081406641526866748145179, −4.94326948027145974159188483948, −3.64122276672664329547582642083, −3.14189296529210666724219205772, −1.46346710718964499944278432316, 0,
1.46346710718964499944278432316, 3.14189296529210666724219205772, 3.64122276672664329547582642083, 4.94326948027145974159188483948, 5.47324081406641526866748145179, 6.27013422173559366918615833585, 7.49774877417135048713191737937, 8.203981792922301940504110098742, 8.836974720135206255610544472959
