L(s) = 1 | + (0.939 − 1.62i)7-s + (0.673 + 1.85i)13-s + (−0.5 + 0.866i)19-s + (0.939 − 0.342i)25-s + (1.11 + 0.642i)31-s − 1.28i·37-s + (−0.266 − 1.50i)43-s + (−1.26 − 2.19i)49-s + (−0.0603 + 0.342i)61-s + (−0.439 + 0.524i)67-s + (−0.326 − 0.118i)73-s + (0.233 − 0.642i)79-s + (3.64 + 0.642i)91-s + (−1.11 − 1.32i)97-s + (−1.70 + 0.984i)103-s + ⋯ |
L(s) = 1 | + (0.939 − 1.62i)7-s + (0.673 + 1.85i)13-s + (−0.5 + 0.866i)19-s + (0.939 − 0.342i)25-s + (1.11 + 0.642i)31-s − 1.28i·37-s + (−0.266 − 1.50i)43-s + (−1.26 − 2.19i)49-s + (−0.0603 + 0.342i)61-s + (−0.439 + 0.524i)67-s + (−0.326 − 0.118i)73-s + (0.233 − 0.642i)79-s + (3.64 + 0.642i)91-s + (−1.11 − 1.32i)97-s + (−1.70 + 0.984i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.419796788\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.419796788\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 1.28iT - T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.233 + 0.642i)T + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \) |
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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
−8.755677205042851612416359342992, −8.326435740993246523863298795078, −7.27341200647528206133582853837, −6.90257710591046892379693291440, −6.01391715720720811382677776282, −4.78303606592705515665356501300, −4.24668603952314352125614261726, −3.60817876515959616251396104805, −2.03357108038411536476503018909, −1.17747433708757185479128214790,
1.27074827134321347631009450844, 2.58612481982095659005971066799, 3.08927906755982887035253645996, 4.57819724793505743885516438634, 5.19219817562293117150937858841, 5.89192992367158226041859140280, 6.60343589740727541380990051893, 7.925001884022442366157726085755, 8.232115303200598623719385776786, 8.878181913091904374681772333832