| L(s) = 1 | + (3.87 + i)2-s + 6i·3-s + (14.0 + 7.74i)4-s + (−6 + 23.2i)6-s + 61.9·7-s + (46.4 + 44.0i)8-s + 45·9-s − 26·11-s + (−46.4 + 84.0i)12-s − 30.9·13-s + (240. + 61.9i)14-s + (136. + 216. i)16-s − 226i·17-s + (174. + 45i)18-s − 134·19-s + ⋯ |
| L(s) = 1 | + (0.968 + 0.250i)2-s + 0.666i·3-s + (0.875 + 0.484i)4-s + (−0.166 + 0.645i)6-s + 1.26·7-s + (0.726 + 0.687i)8-s + 0.555·9-s − 0.214·11-s + (−0.322 + 0.583i)12-s − 0.183·13-s + (1.22 + 0.316i)14-s + (0.531 + 0.847i)16-s − 0.782i·17-s + (0.537 + 0.138i)18-s − 0.371·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.173639777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.173639777\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.87 - i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 6iT - 81T^{2} \) |
| 7 | \( 1 - 61.9T + 2.40e3T^{2} \) |
| 11 | \( 1 + 26T + 1.46e4T^{2} \) |
| 13 | \( 1 + 30.9T + 2.85e4T^{2} \) |
| 17 | \( 1 + 226iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 134T + 1.30e5T^{2} \) |
| 23 | \( 1 - 309.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 340. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.23e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.76e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 994T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.88e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.10e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.81e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.01e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.07e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 8.00e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 557. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 386iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 1.10e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 2.23e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.00e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + 8.73e3iT - 8.85e7T^{2} \) |
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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
−12.03954223060210339248861192045, −11.06937897642603494105924246832, −10.37419878458310276061682475529, −8.901025171352311627724465432495, −7.73579549375819375336936890880, −6.79669656794126128244254138833, −5.13328548596354668574856748663, −4.74081806758853543109641312924, −3.39301880949472645150760173841, −1.76687287044139445092573516888,
1.26888327770582587443287127240, 2.28309233350093321790852976424, 4.05846434746697880192124241781, 5.05594846720597200797086529005, 6.28525317830759578009162285909, 7.37683841794104547382776463487, 8.221474875218072421039479016074, 9.927411807671134985097552981644, 10.96870667018226851299817214936, 11.70867160837334694494337105991
