L(s) = 1 | + (−0.327 + 0.945i)2-s + (−0.653 + 1.43i)3-s + (−0.786 − 0.618i)4-s + (−0.959 − 0.281i)5-s + (−1.13 − 1.08i)6-s + (−1.95 + 0.376i)7-s + (0.841 − 0.540i)8-s + (−0.963 − 1.11i)9-s + (0.580 − 0.814i)10-s + (1.39 − 0.720i)12-s + (0.283 − 1.97i)14-s + (1.02 − 1.18i)15-s + (0.235 + 0.971i)16-s + (1.36 − 0.546i)18-s + (0.580 + 0.814i)20-s + (0.737 − 3.04i)21-s + ⋯ |
L(s) = 1 | + (−0.327 + 0.945i)2-s + (−0.653 + 1.43i)3-s + (−0.786 − 0.618i)4-s + (−0.959 − 0.281i)5-s + (−1.13 − 1.08i)6-s + (−1.95 + 0.376i)7-s + (0.841 − 0.540i)8-s + (−0.963 − 1.11i)9-s + (0.580 − 0.814i)10-s + (1.39 − 0.720i)12-s + (0.283 − 1.97i)14-s + (1.02 − 1.18i)15-s + (0.235 + 0.971i)16-s + (1.36 − 0.546i)18-s + (0.580 + 0.814i)20-s + (0.737 − 3.04i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1377745596\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1377745596\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.327 - 0.945i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 67 | \( 1 + (-0.928 + 0.371i)T \) |
good | 3 | \( 1 + (0.653 - 1.43i)T + (-0.654 - 0.755i)T^{2} \) |
| 7 | \( 1 + (1.95 - 0.376i)T + (0.928 - 0.371i)T^{2} \) |
| 11 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 13 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 17 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 19 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 23 | \( 1 + (-0.995 + 0.0950i)T + (0.981 - 0.189i)T^{2} \) |
| 29 | \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.0883 - 0.0353i)T + (0.723 + 0.690i)T^{2} \) |
| 43 | \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \) |
| 47 | \( 1 + (1.03 + 1.44i)T + (-0.327 + 0.945i)T^{2} \) |
| 53 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 61 | \( 1 + (1.38 + 1.32i)T + (0.0475 + 0.998i)T^{2} \) |
| 71 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 73 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 79 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 83 | \( 1 + (-0.462 - 1.90i)T + (-0.888 + 0.458i)T^{2} \) |
| 89 | \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.593658782464571322590117522348, −9.166118042712494610820021244529, −8.452544443047057580657254683581, −7.13282040866237937191876764618, −6.57110066048506986023398814264, −5.53801953140480117283424215960, −4.97744813466369644834039999737, −3.85566648335290344004003635723, −3.34985294253609396317227837812, −0.17551713050757869359012690629,
1.03652046509493954394949484314, 2.67896435307456434736608912155, 3.37379193052877670450676378542, 4.45987914096870670961000381854, 5.94320079633574648342366671223, 6.70236039537464077944419039683, 7.38606005802106352678210454287, 8.002949254904411352968679868717, 9.148635914178771256733006237763, 9.870218796120561462538956603808