L(s) = 1 | + (−0.327 + 0.945i)3-s + (0.959 + 0.281i)4-s + (1.07 + 1.36i)7-s + (−0.786 − 0.618i)9-s + (−0.580 + 0.814i)12-s + (0.184 − 0.0448i)13-s + (0.841 + 0.540i)16-s + (−1.65 − 0.850i)19-s + (−1.63 + 0.566i)21-s + (−0.235 − 0.971i)25-s + (0.841 − 0.540i)27-s + (0.643 + 1.60i)28-s + (1.30 − 1.50i)31-s + (−0.580 − 0.814i)36-s + (−1.65 − 0.318i)37-s + ⋯ |
L(s) = 1 | + (−0.327 + 0.945i)3-s + (0.959 + 0.281i)4-s + (1.07 + 1.36i)7-s + (−0.786 − 0.618i)9-s + (−0.580 + 0.814i)12-s + (0.184 − 0.0448i)13-s + (0.841 + 0.540i)16-s + (−1.65 − 0.850i)19-s + (−1.63 + 0.566i)21-s + (−0.235 − 0.971i)25-s + (0.841 − 0.540i)27-s + (0.643 + 1.60i)28-s + (1.30 − 1.50i)31-s + (−0.580 − 0.814i)36-s + (−1.65 − 0.318i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.252866613\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.252866613\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.327 - 0.945i)T \) |
| 397 | \( 1 - T \) |
good | 2 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 5 | \( 1 + (0.235 + 0.971i)T^{2} \) |
| 7 | \( 1 + (-1.07 - 1.36i)T + (-0.235 + 0.971i)T^{2} \) |
| 11 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 13 | \( 1 + (-0.184 + 0.0448i)T + (0.888 - 0.458i)T^{2} \) |
| 17 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (1.65 + 0.850i)T + (0.580 + 0.814i)T^{2} \) |
| 23 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 29 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 31 | \( 1 + (-1.30 + 1.50i)T + (-0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (1.65 + 0.318i)T + (0.928 + 0.371i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.223 - 1.55i)T + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 53 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (-0.786 + 0.618i)T^{2} \) |
| 61 | \( 1 + (1.83 + 0.445i)T + (0.888 + 0.458i)T^{2} \) |
| 67 | \( 1 + (-1.88 - 0.363i)T + (0.928 + 0.371i)T^{2} \) |
| 71 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (-0.0224 + 0.470i)T + (-0.995 - 0.0950i)T^{2} \) |
| 79 | \( 1 + (0.327 + 0.566i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (-0.888 - 0.458i)T^{2} \) |
| 97 | \( 1 + (0.0883 + 0.0353i)T + (0.723 + 0.690i)T^{2} \) |
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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
−10.35574799076323150264011168548, −9.272163549248563404366900496830, −8.465240673602050371975478500741, −7.986985940263739280952233707285, −6.49939522644599150389057372467, −6.02226350614866112159679931195, −5.00567453540514632742309915646, −4.20923227622396089097335357297, −2.84042389137820108258507608783, −2.07024043898759107144297437644,
1.30482457263549992950813514192, 1.99986209628792604472648204882, 3.49520277341051445797180795672, 4.75654836902816376914627666272, 5.69676202345955855029697694832, 6.73715511353745862632454544426, 7.08392409550013640085476796018, 8.010430719084535199923371009734, 8.552155804757439432737177387616, 10.28115414678735816553445696372