L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 4·6-s + 3·7-s + 9-s + 4·12-s + 4·13-s − 6·14-s − 4·16-s + 17-s − 2·18-s − 2·19-s + 6·21-s − 2·23-s − 8·26-s − 4·27-s + 6·28-s + 2·29-s − 2·31-s + 8·32-s − 2·34-s + 2·36-s + 2·37-s + 4·38-s + 8·39-s + 5·41-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 1.63·6-s + 1.13·7-s + 1/3·9-s + 1.15·12-s + 1.10·13-s − 1.60·14-s − 16-s + 0.242·17-s − 0.471·18-s − 0.458·19-s + 1.30·21-s − 0.417·23-s − 1.56·26-s − 0.769·27-s + 1.13·28-s + 0.371·29-s − 0.359·31-s + 1.41·32-s − 0.342·34-s + 1/3·36-s + 0.328·37-s + 0.648·38-s + 1.28·39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.78156632921751, −14.30265670024787, −13.81306343506847, −13.33257532839737, −12.88390346718824, −11.82560120208223, −11.59676439170086, −10.92808916880716, −10.53289616171853, −9.984037871006801, −9.364857762256033, −8.824852633428501, −8.553895917928944, −8.098801897357719, −7.712359078327562, −7.191510946429351, −6.415524845385156, −5.786772982007685, −4.979570176217247, −4.254909135547861, −3.758813045005936, −2.888077457066762, −2.254603430164585, −1.594197030294226, −1.156287009535381, 0,
1.156287009535381, 1.594197030294226, 2.254603430164585, 2.888077457066762, 3.758813045005936, 4.254909135547861, 4.979570176217247, 5.786772982007685, 6.415524845385156, 7.191510946429351, 7.712359078327562, 8.098801897357719, 8.553895917928944, 8.824852633428501, 9.364857762256033, 9.984037871006801, 10.53289616171853, 10.92808916880716, 11.59676439170086, 11.82560120208223, 12.88390346718824, 13.33257532839737, 13.81306343506847, 14.30265670024787, 14.78156632921751