| L(s) = 1 | − 3-s + 3.70i·5-s + 3.23·7-s + 9-s − 6.47·11-s − 1.74i·13-s − 3.70i·15-s + 6.53i·17-s + 4.57i·19-s − 3.23·21-s − 7.19i·23-s − 8.70·25-s − 27-s + 4.78i·29-s + 6.32i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.65i·5-s + 1.22·7-s + 0.333·9-s − 1.95·11-s − 0.484i·13-s − 0.955i·15-s + 1.58i·17-s + 1.04i·19-s − 0.706·21-s − 1.50i·23-s − 1.74·25-s − 0.192·27-s + 0.888i·29-s + 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.562529 + 0.827241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.562529 + 0.827241i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 + (-2.23 - 5.65i)T \) |
| good | 5 | \( 1 - 3.70iT - 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 + 6.47T + 11T^{2} \) |
| 13 | \( 1 + 1.74iT - 13T^{2} \) |
| 17 | \( 1 - 6.53iT - 17T^{2} \) |
| 19 | \( 1 - 4.57iT - 19T^{2} \) |
| 23 | \( 1 + 7.19iT - 23T^{2} \) |
| 29 | \( 1 - 4.78iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 + 0.874iT - 59T^{2} \) |
| 61 | \( 1 + 5.65iT - 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 7.19iT - 89T^{2} \) |
| 97 | \( 1 - 5.24iT - 97T^{2} \) |
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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
−10.89914865219034375249851063031, −10.66439464691099722690154634525, −10.23434521442857884465072617011, −8.226718039227588853808239428241, −7.84278729022676306832511203184, −6.69309701469450201278367063601, −5.76411427351526218026164123328, −4.78458778129742216141964141900, −3.29315263553553006769799925086, −2.06526684004951562104115951224,
0.66836703962159163295753653542, 2.25518090008552879901056756309, 4.45705201642717063481790554703, 5.06406068454349665520706912871, 5.60720483243913277086941191967, 7.48875327155824554270782721120, 7.943292877590901717091025525057, 9.082593527901776626273862153902, 9.798196654489305923631162091175, 11.25190755246514303710826099304
