| L(s) = 1 | − 2-s + 2·3-s + 4-s − 3·5-s − 2·6-s − 3·8-s + 3·9-s + 3·10-s − 11-s + 2·12-s − 5·13-s − 6·15-s + 16-s − 2·17-s − 3·18-s − 3·19-s − 3·20-s + 22-s − 9·23-s − 6·24-s + 25-s + 5·26-s + 4·27-s + 6·30-s + 2·31-s + 32-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 1.34·5-s − 0.816·6-s − 1.06·8-s + 9-s + 0.948·10-s − 0.301·11-s + 0.577·12-s − 1.38·13-s − 1.54·15-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.688·19-s − 0.670·20-s + 0.213·22-s − 1.87·23-s − 1.22·24-s + 1/5·25-s + 0.980·26-s + 0.769·27-s + 1.09·30-s + 0.359·31-s + 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6245001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6245001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 226 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.619968132065099892177203729138, −8.316003717697465366930256121178, −7.972774542440843912392888129446, −7.75145885241878359648895542211, −7.46707527743507279578913977357, −6.87905813560172738543832622505, −6.69677117957831974543849297954, −6.19178460097461128428932670083, −5.64861106320863544774328735532, −5.14472615012531601281039040470, −4.53741264511365346429213094176, −4.17846327595102943021348064091, −3.82646673543101944121250915334, −3.47090093139165691657595265246, −2.64543960302450094173766878465, −2.40564081120874412582855903903, −2.20557450218349649384417950242, −1.20349198915179027046356442468, 0, 0,
1.20349198915179027046356442468, 2.20557450218349649384417950242, 2.40564081120874412582855903903, 2.64543960302450094173766878465, 3.47090093139165691657595265246, 3.82646673543101944121250915334, 4.17846327595102943021348064091, 4.53741264511365346429213094176, 5.14472615012531601281039040470, 5.64861106320863544774328735532, 6.19178460097461128428932670083, 6.69677117957831974543849297954, 6.87905813560172738543832622505, 7.46707527743507279578913977357, 7.75145885241878359648895542211, 7.972774542440843912392888129446, 8.316003717697465366930256121178, 8.619968132065099892177203729138
