| L(s) = 1 | − 3·2-s − 7·3-s + 4-s + 5·5-s + 21·6-s + 11·7-s + 21·8-s + 22·9-s − 15·10-s − 36·11-s − 7·12-s − 65·13-s − 33·14-s − 35·15-s − 71·16-s − 87·17-s − 66·18-s + 5·20-s − 77·21-s + 108·22-s − 129·23-s − 147·24-s + 25·25-s + 195·26-s + 35·27-s + 11·28-s − 231·29-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1.34·3-s + 1/8·4-s + 0.447·5-s + 1.42·6-s + 0.593·7-s + 0.928·8-s + 0.814·9-s − 0.474·10-s − 0.986·11-s − 0.168·12-s − 1.38·13-s − 0.629·14-s − 0.602·15-s − 1.10·16-s − 1.24·17-s − 0.864·18-s + 0.0559·20-s − 0.800·21-s + 1.04·22-s − 1.16·23-s − 1.25·24-s + 1/5·25-s + 1.47·26-s + 0.249·27-s + 0.0742·28-s − 1.47·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - p T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 11 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 87 T + p^{3} T^{2} \) |
| 23 | \( 1 + 129 T + p^{3} T^{2} \) |
| 29 | \( 1 + 231 T + p^{3} T^{2} \) |
| 31 | \( 1 + 110 T + p^{3} T^{2} \) |
| 37 | \( 1 - 142 T + p^{3} T^{2} \) |
| 41 | \( 1 - 330 T + p^{3} T^{2} \) |
| 43 | \( 1 - 74 T + p^{3} T^{2} \) |
| 47 | \( 1 + 336 T + p^{3} T^{2} \) |
| 53 | \( 1 + 501 T + p^{3} T^{2} \) |
| 59 | \( 1 + 633 T + p^{3} T^{2} \) |
| 61 | \( 1 + 88 T + p^{3} T^{2} \) |
| 67 | \( 1 + 119 T + p^{3} T^{2} \) |
| 71 | \( 1 - 204 T + p^{3} T^{2} \) |
| 73 | \( 1 - 407 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1262 T + p^{3} T^{2} \) |
| 83 | \( 1 - 270 T + p^{3} T^{2} \) |
| 89 | \( 1 - 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1406 T + p^{3} T^{2} \) |
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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
−7.976104875040179980625670119065, −7.55570377165230706586541769789, −6.58205448509320717034944830437, −5.65524938186365926058594697249, −4.94166243505743151117467079197, −4.36142709963211582738873967710, −2.42934013961845190971299385390, −1.56824912642992177401339934454, 0, 0,
1.56824912642992177401339934454, 2.42934013961845190971299385390, 4.36142709963211582738873967710, 4.94166243505743151117467079197, 5.65524938186365926058594697249, 6.58205448509320717034944830437, 7.55570377165230706586541769789, 7.976104875040179980625670119065
