| L(s) = 1 | + 3.02·3-s + i·5-s + (2.17 + 1.51i)7-s + 6.12·9-s + 4.71i·11-s + 2i·13-s + 3.02i·15-s − 1.12i·17-s − 4.71·19-s + (6.56 + 4.56i)21-s − 6.41i·23-s − 25-s + 9.43·27-s + 2·29-s + 3.39·31-s + ⋯ |
| L(s) = 1 | + 1.74·3-s + 0.447i·5-s + (0.821 + 0.570i)7-s + 2.04·9-s + 1.42i·11-s + 0.554i·13-s + 0.779i·15-s − 0.272i·17-s − 1.08·19-s + (1.43 + 0.995i)21-s − 1.33i·23-s − 0.200·25-s + 1.81·27-s + 0.371·29-s + 0.609·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.667889724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.667889724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.17 - 1.51i)T \) |
| good | 3 | \( 1 - 3.02T + 3T^{2} \) |
| 11 | \( 1 - 4.71iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 1.12iT - 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 + 6.41iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 1.12iT - 41T^{2} \) |
| 43 | \( 1 + 0.371iT - 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 2.06T + 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + 3.76iT - 67T^{2} \) |
| 71 | \( 1 - 7.36iT - 71T^{2} \) |
| 73 | \( 1 + 15.3iT - 73T^{2} \) |
| 79 | \( 1 + 1.32iT - 79T^{2} \) |
| 83 | \( 1 - 3.02T + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 1.12iT - 97T^{2} \) |
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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
−8.917352123551799742355701933176, −8.503228546186306248968372653915, −7.74058948000157341598117268422, −7.07117162857781606592001655806, −6.29218769819815041274161026027, −4.65015497723044100814226302873, −4.42623882118067587718990106429, −3.15611737980817596938762478897, −2.25799575551080620138605966327, −1.82986736809007935956284936199,
1.06543700321972732572875287819, 2.06833822270701356252698972111, 3.18206314759799960520684925133, 3.80363489606849949036209430998, 4.67261323334000268335572399582, 5.70733129556746101263981875049, 6.82405049941678640303462388874, 7.82358485377905719572435458176, 8.203776603609088519778350255245, 8.673555086055850448316711179879
