L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.326 + 1.85i)3-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)6-s + (−0.500 + 0.866i)8-s + (−2.37 − 0.866i)9-s + (−0.173 + 0.300i)11-s + (0.939 + 1.62i)12-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 2.53·18-s + (0.766 − 0.642i)19-s + (0.0603 − 0.342i)22-s + (−1.43 − 1.20i)24-s + (0.173 + 0.984i)25-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.326 + 1.85i)3-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)6-s + (−0.500 + 0.866i)8-s + (−2.37 − 0.866i)9-s + (−0.173 + 0.300i)11-s + (0.939 + 1.62i)12-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 2.53·18-s + (0.766 − 0.642i)19-s + (0.0603 − 0.342i)22-s + (−1.43 − 1.20i)24-s + (0.173 + 0.984i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3974292175\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3974292175\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (-0.766 + 0.642i)T \) |
good | 3 | \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
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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
−14.02535236977326069349586725758, −11.99817812850752840166902379306, −11.12773911450365579764331666597, −10.30210052813183256643901398852, −9.537530903351788838902073427769, −8.818110884497844507019730665502, −7.39758519646285917014352048049, −5.78546712841777911210077602306, −4.91519720797899269806536893658, −3.20332236583521029241306119119,
1.39037118027808155937599020260, 2.93868752547714674847099105795, 5.80589266530993807496984830934, 6.77135032788205920240945245072, 7.83150068134229021638421772963, 8.364272804129913652850256181892, 9.906150846988889769290051670243, 11.15418693766925744628716813408, 11.97180342685846314469107451509, 12.61827844700886109987831941579