L(s) = 1 | + (0.5 − 0.866i)2-s + (3.5 + 6.06i)4-s + (2.5 + 4.33i)5-s + (12 − 20.7i)7-s + 15·8-s + 5·10-s + (26 − 45.0i)11-s + (−11 − 19.0i)13-s + (−12 − 20.7i)14-s + (−20.5 + 35.5i)16-s + 14·17-s − 20·19-s + (−17.5 + 30.3i)20-s + (−25.9 − 45.0i)22-s + (−84 − 145. i)23-s + ⋯ |
L(s) = 1 | + (0.176 − 0.306i)2-s + (0.437 + 0.757i)4-s + (0.223 + 0.387i)5-s + (0.647 − 1.12i)7-s + 0.662·8-s + 0.158·10-s + (0.712 − 1.23i)11-s + (−0.234 − 0.406i)13-s + (−0.229 − 0.396i)14-s + (−0.320 + 0.554i)16-s + 0.199·17-s − 0.241·19-s + (−0.195 + 0.338i)20-s + (−0.251 − 0.436i)22-s + (−0.761 − 1.31i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.762215582\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.762215582\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-12 + 20.7i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-26 + 45.0i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (11 + 19.0i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 14T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20T + 6.85e3T^{2} \) |
| 23 | \( 1 + (84 + 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-115 + 199. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-144 - 249. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 34T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-61 - 105. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-94 + 162. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-128 + 221. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 338T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-50 - 86.6i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (371 - 642. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-42 - 72.7i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 328T + 3.57e5T^{2} \) |
| 73 | \( 1 + 38T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-120 + 207. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-606 + 1.04e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 330T + 7.04e5T^{2} \) |
| 97 | \( 1 + (433 - 749. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.65493064884242599161639351004, −10.32232539591604636642634425035, −8.651110492577223436203193731980, −7.981751833576994079897342614293, −6.99534826810355526992144597728, −6.13550299264119305668137282702, −4.52249342866656905974037308526, −3.65351362625937442271321541951, −2.51622618802416146559391600427, −0.953746729995841975647049243499,
1.46225288388901041440202629709, 2.28047480859340653055196837158, 4.34906766025708677513403836662, 5.22316394867006171987224045607, 6.05969318382506731939254356349, 7.06183772594267553291840387247, 8.105584672680776701064768573066, 9.330306917979044149728506267848, 9.788485091838402497646613926602, 11.03685191165815487576769240091