| L(s) = 1 | + 0.398i·2-s + 1.84·4-s + 5-s + 1.52i·8-s + 0.398i·10-s − 5.13i·11-s + 5.09i·13-s + 3.07·16-s + 4.48·17-s − 3.80i·19-s + 1.84·20-s + 2.04·22-s + 2.64i·23-s + 25-s − 2.02·26-s + ⋯ |
| L(s) = 1 | + 0.281i·2-s + 0.920·4-s + 0.447·5-s + 0.540i·8-s + 0.125i·10-s − 1.54i·11-s + 1.41i·13-s + 0.768·16-s + 1.08·17-s − 0.872i·19-s + 0.411·20-s + 0.435·22-s + 0.551i·23-s + 0.200·25-s − 0.397·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.637302552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.637302552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.398iT - 2T^{2} \) |
| 11 | \( 1 + 5.13iT - 11T^{2} \) |
| 13 | \( 1 - 5.09iT - 13T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 + 3.80iT - 19T^{2} \) |
| 23 | \( 1 - 2.64iT - 23T^{2} \) |
| 29 | \( 1 + 9.13iT - 29T^{2} \) |
| 31 | \( 1 - 4.75iT - 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 - 7.51T + 41T^{2} \) |
| 43 | \( 1 - 6.73T + 43T^{2} \) |
| 47 | \( 1 - 7.82T + 47T^{2} \) |
| 53 | \( 1 + 2.89iT - 53T^{2} \) |
| 59 | \( 1 - 4.93T + 59T^{2} \) |
| 61 | \( 1 - 11.8iT - 61T^{2} \) |
| 67 | \( 1 - 6.99T + 67T^{2} \) |
| 71 | \( 1 + 1.23iT - 71T^{2} \) |
| 73 | \( 1 + 0.578iT - 73T^{2} \) |
| 79 | \( 1 + 4.93T + 79T^{2} \) |
| 83 | \( 1 + 9.75T + 83T^{2} \) |
| 89 | \( 1 + 7.27T + 89T^{2} \) |
| 97 | \( 1 - 0.313iT - 97T^{2} \) |
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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
−8.972433649236711568052292303223, −8.329655693760264023020089451141, −7.36399748007477047574151257757, −6.77791974985075585579512835885, −5.86694582743422186136978325873, −5.52472574878728576185120811779, −4.16582196429700684627358427840, −3.13757670716876417625631698697, −2.27211629477578584440012588267, −1.09835152869597991041355818255,
1.16972325354135443536433786395, 2.15408371492349027729242108777, 3.02315771284286804213086785014, 3.97007318376207284663961213671, 5.26050189248667022820933456818, 5.77362579700081872597143340418, 6.79685995658971781996766930250, 7.46048337042584291992714719686, 8.037846072586456032549917244969, 9.202799610553550834300208133163
