L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 21·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 42·10-s − 11·11-s − 12·12-s + 65·13-s − 14·14-s + 63·15-s + 16·16-s − 54·17-s − 18·18-s + 65·19-s − 84·20-s − 21·21-s + 22·22-s + 132·23-s + 24·24-s + 316·25-s − 130·26-s − 27·27-s + 28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.87·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.32·10-s − 0.301·11-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 1.08·15-s + 1/4·16-s − 0.770·17-s − 0.235·18-s + 0.784·19-s − 0.939·20-s − 0.218·21-s + 0.213·22-s + 1.19·23-s + 0.204·24-s + 2.52·25-s − 0.980·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
good | 5 | \( 1 + 21 T + p^{3} T^{2} \) |
| 13 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 65 T + p^{3} T^{2} \) |
| 23 | \( 1 - 132 T + p^{3} T^{2} \) |
| 29 | \( 1 - 39 T + p^{3} T^{2} \) |
| 31 | \( 1 + 178 T + p^{3} T^{2} \) |
| 37 | \( 1 + 439 T + p^{3} T^{2} \) |
| 41 | \( 1 - 96 T + p^{3} T^{2} \) |
| 43 | \( 1 - 272 T + p^{3} T^{2} \) |
| 47 | \( 1 + 375 T + p^{3} T^{2} \) |
| 53 | \( 1 - 612 T + p^{3} T^{2} \) |
| 59 | \( 1 + 507 T + p^{3} T^{2} \) |
| 61 | \( 1 - 758 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1087 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 673 T + p^{3} T^{2} \) |
| 79 | \( 1 + 700 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1218 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1350 T + p^{3} T^{2} \) |
| 97 | \( 1 + 808 T + p^{3} T^{2} \) |
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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
−10.59166568389414082778525296331, −8.982474775953486935120096084621, −8.419904224089616275910891363335, −7.43837587273778303753974332021, −6.84461666751090078344700866005, −5.41033251951790938584064553040, −4.23374206903403509779299139841, −3.22857464346595532087413370523, −1.19025371572973548094607805642, 0,
1.19025371572973548094607805642, 3.22857464346595532087413370523, 4.23374206903403509779299139841, 5.41033251951790938584064553040, 6.84461666751090078344700866005, 7.43837587273778303753974332021, 8.419904224089616275910891363335, 8.982474775953486935120096084621, 10.59166568389414082778525296331