L(s) = 1 | + (−0.916 − 1.46i)3-s − 0.860i·5-s + 2.31i·7-s + (−1.31 + 2.69i)9-s − 0.512·11-s + 2.93·13-s + (−1.26 + 0.789i)15-s + 5.22i·17-s + i·19-s + (3.40 − 2.12i)21-s − 9.29·23-s + 4.25·25-s + (5.16 − 0.530i)27-s + 6.87i·29-s + 4.93i·31-s + ⋯ |
L(s) = 1 | + (−0.529 − 0.848i)3-s − 0.385i·5-s + 0.876i·7-s + (−0.439 + 0.898i)9-s − 0.154·11-s + 0.815·13-s + (−0.326 + 0.203i)15-s + 1.26i·17-s + 0.229i·19-s + (0.743 − 0.463i)21-s − 1.93·23-s + 0.851·25-s + (0.994 − 0.102i)27-s + 1.27i·29-s + 0.887i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07061 + 0.306517i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07061 + 0.306517i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.916 + 1.46i)T \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 + 0.860iT - 5T^{2} \) |
| 7 | \( 1 - 2.31iT - 7T^{2} \) |
| 11 | \( 1 + 0.512T + 11T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 - 5.22iT - 17T^{2} \) |
| 23 | \( 1 + 9.29T + 23T^{2} \) |
| 29 | \( 1 - 6.87iT - 29T^{2} \) |
| 31 | \( 1 - 4.93iT - 31T^{2} \) |
| 37 | \( 1 - 9.45T + 37T^{2} \) |
| 41 | \( 1 + 6.43iT - 41T^{2} \) |
| 43 | \( 1 - 1.68iT - 43T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 - 3.31iT - 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 + 2.01T + 61T^{2} \) |
| 67 | \( 1 + 7.15iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.49T + 73T^{2} \) |
| 79 | \( 1 - 5.36iT - 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 - 1.04iT - 89T^{2} \) |
| 97 | \( 1 + 8.21T + 97T^{2} \) |
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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
−10.43167345641327600438514735136, −9.088667558623994901935383187683, −8.412502618871531625793106647184, −7.76305664491729000305866019023, −6.52103751488548994735768900951, −5.93000122105451061377599139543, −5.16776455886503541342703161581, −3.86773212711134859872092540824, −2.39648156618768980761261452535, −1.31296993488135585552825735991,
0.62557752639124209564220802831, 2.68368087685289178278918029702, 3.90588777145161675945154334781, 4.48988056655494588840916919656, 5.72486270657640074575797433698, 6.42899259679481667866371921521, 7.44366510804420419633444775707, 8.352750986725270098625813547465, 9.549476658144131500307828562681, 9.971336621330989588954863604220