L(s) = 1 | + (0.895 − 0.445i)2-s + (0.602 − 0.798i)4-s + (−0.932 + 1.36i)5-s + (0.183 − 0.982i)8-s + (0.673 − 0.739i)9-s + (−0.227 + 1.63i)10-s + (1.32 + 0.247i)13-s + (−0.273 − 0.961i)16-s + (0.273 + 0.961i)17-s + (0.273 − 0.961i)18-s + (0.524 + 1.56i)20-s + (−0.622 − 1.60i)25-s + (1.29 − 0.368i)26-s + (0.0127 + 0.0914i)29-s + (−0.673 − 0.739i)32-s + ⋯ |
L(s) = 1 | + (0.895 − 0.445i)2-s + (0.602 − 0.798i)4-s + (−0.932 + 1.36i)5-s + (0.183 − 0.982i)8-s + (0.673 − 0.739i)9-s + (−0.227 + 1.63i)10-s + (1.32 + 0.247i)13-s + (−0.273 − 0.961i)16-s + (0.273 + 0.961i)17-s + (0.273 − 0.961i)18-s + (0.524 + 1.56i)20-s + (−0.622 − 1.60i)25-s + (1.29 − 0.368i)26-s + (0.0127 + 0.0914i)29-s + (−0.673 − 0.739i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.646236614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646236614\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.895 + 0.445i)T \) |
| 17 | \( 1 + (-0.273 - 0.961i)T \) |
good | 3 | \( 1 + (-0.673 + 0.739i)T^{2} \) |
| 5 | \( 1 + (0.932 - 1.36i)T + (-0.361 - 0.932i)T^{2} \) |
| 7 | \( 1 + (0.995 - 0.0922i)T^{2} \) |
| 11 | \( 1 + (-0.961 + 0.273i)T^{2} \) |
| 13 | \( 1 + (-1.32 - 0.247i)T + (0.932 + 0.361i)T^{2} \) |
| 19 | \( 1 + (-0.602 - 0.798i)T^{2} \) |
| 23 | \( 1 + (0.995 - 0.0922i)T^{2} \) |
| 29 | \( 1 + (-0.0127 - 0.0914i)T + (-0.961 + 0.273i)T^{2} \) |
| 31 | \( 1 + (-0.361 + 0.932i)T^{2} \) |
| 37 | \( 1 + (0.618 - 0.145i)T + (0.895 - 0.445i)T^{2} \) |
| 41 | \( 1 + (0.456 + 1.03i)T + (-0.673 + 0.739i)T^{2} \) |
| 43 | \( 1 + (-0.850 - 0.526i)T^{2} \) |
| 47 | \( 1 + (-0.0922 + 0.995i)T^{2} \) |
| 53 | \( 1 + (1.34 - 1.47i)T + (-0.0922 - 0.995i)T^{2} \) |
| 59 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
| 61 | \( 1 + (-0.516 + 0.621i)T + (-0.183 - 0.982i)T^{2} \) |
| 67 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 71 | \( 1 + (-0.995 + 0.0922i)T^{2} \) |
| 73 | \( 1 + (1.74 + 0.972i)T + (0.526 + 0.850i)T^{2} \) |
| 79 | \( 1 + (-0.798 + 0.602i)T^{2} \) |
| 83 | \( 1 + (0.739 - 0.673i)T^{2} \) |
| 89 | \( 1 + (1.75 - 0.328i)T + (0.932 - 0.361i)T^{2} \) |
| 97 | \( 1 + (-1.97 + 0.0914i)T + (0.995 - 0.0922i)T^{2} \) |
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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
−10.43632423829110865465069173910, −9.342615021835964733759838589906, −8.155516269919666853379717139838, −7.16020132794589624250630156578, −6.54543116135613456719067293533, −5.88903041267553251071255457482, −4.38666743188274327178785696444, −3.66797133756073618766362552959, −3.14405107054416629226301574436, −1.60048085806193551603809729281,
1.51000763534474136452288407846, 3.21666784931427297528130653947, 4.15874793288860255845578195902, 4.83975744246636563633137723045, 5.52385151017163762015439824447, 6.70148763308032384453495113584, 7.63881718335639626743203321106, 8.212345972393903794750878246408, 8.853069664661217198807650120497, 10.05459564292355226428535025774