| L(s) = 1 | + (−11.5 + 11.0i)2-s − 27.2i·3-s + (11.6 − 255. i)4-s + (301. + 315. i)6-s + 3.32e3i·7-s + (2.69e3 + 3.08e3i)8-s + 5.81e3·9-s − 6.36e3i·11-s + (−6.96e3 − 316. i)12-s + 3.07e4·13-s + (−3.67e4 − 3.84e4i)14-s + (−6.52e4 − 5.94e3i)16-s − 1.22e5·17-s + (−6.73e4 + 6.43e4i)18-s + 7.55e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.722 + 0.690i)2-s − 0.336i·3-s + (0.0453 − 0.998i)4-s + (0.232 + 0.243i)6-s + 1.38i·7-s + (0.657 + 0.753i)8-s + 0.886·9-s − 0.434i·11-s + (−0.335 − 0.0152i)12-s + 1.07·13-s + (−0.956 − 1.00i)14-s + (−0.995 − 0.0906i)16-s − 1.46·17-s + (−0.641 + 0.612i)18-s + 0.0579i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0453 - 0.998i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0453 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.994506 + 0.950345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.994506 + 0.950345i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.5 - 11.0i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 27.2iT - 6.56e3T^{2} \) |
| 7 | \( 1 - 3.32e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 6.36e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.07e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.22e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 7.55e3iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 4.06e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 8.28e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.39e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.86e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 9.72e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.61e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 1.87e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.01e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 6.18e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 9.79e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.19e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.62e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.35e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 1.22e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 6.96e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.58e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 5.97e7T + 7.83e15T^{2} \) |
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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
−12.57111829825315130121848622020, −11.29998757605888916319987840332, −10.23870845513598104719014823531, −8.828173539545481986972829063645, −8.432885588270249521109075691182, −6.79786459988678177028836606385, −6.08926423100352639771631533607, −4.66011491981385187530418879858, −2.40504116674588441428504929629, −1.07583856694949742786361108296,
0.60606477858858405659166525140, 1.80026993330052525792997354118, 3.68574615421502994648260971096, 4.41249750713626829546791844388, 6.75693681580545006699568630382, 7.63831818847471751967962230187, 8.976756938154305780142837833180, 10.05016980653358302008581063662, 10.72348433065405962433852267899, 11.68180992052643070208821355943
