| L(s) = 1 | − 2-s + 4-s + (1.67 − 1.47i)5-s − 8-s + (−1.67 + 1.47i)10-s − 6.33i·11-s + 1.05·13-s + 16-s + 4.63i·17-s + 6.17i·19-s + (1.67 − 1.47i)20-s + 6.33i·22-s − 7.04·23-s + (0.624 − 4.96i)25-s − 1.05·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.749 − 0.661i)5-s − 0.353·8-s + (−0.530 + 0.467i)10-s − 1.90i·11-s + 0.292·13-s + 0.250·16-s + 1.12i·17-s + 1.41i·19-s + (0.374 − 0.330i)20-s + 1.34i·22-s − 1.46·23-s + (0.124 − 0.992i)25-s − 0.206·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147470416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147470416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.67 + 1.47i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6.33iT - 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 - 4.63iT - 17T^{2} \) |
| 19 | \( 1 - 6.17iT - 19T^{2} \) |
| 23 | \( 1 + 7.04T + 23T^{2} \) |
| 29 | \( 1 + 2.98iT - 29T^{2} \) |
| 31 | \( 1 + 6.31iT - 31T^{2} \) |
| 37 | \( 1 - 3.47iT - 37T^{2} \) |
| 41 | \( 1 - 6.97T + 41T^{2} \) |
| 43 | \( 1 + 2.58iT - 43T^{2} \) |
| 47 | \( 1 + 8.15iT - 47T^{2} \) |
| 53 | \( 1 - 8.87T + 53T^{2} \) |
| 59 | \( 1 - 0.904T + 59T^{2} \) |
| 61 | \( 1 + 10.1iT - 61T^{2} \) |
| 67 | \( 1 - 9.83iT - 67T^{2} \) |
| 71 | \( 1 + 14.4iT - 71T^{2} \) |
| 73 | \( 1 + 8.36T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 - 3.83T + 89T^{2} \) |
| 97 | \( 1 + 5.87T + 97T^{2} \) |
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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
−8.250426251628324410971290865068, −7.77446096715541953662465396639, −6.37923405323460932372262246673, −5.91481564584721424415152938772, −5.62292587451544366453780678310, −4.15498024692626461528321713912, −3.51659055775236908047035810343, −2.28838897272192755828338034875, −1.46654511804521036706397097683, −0.40688400338634820258337564739,
1.28925041496334317884997137308, 2.31407654722873901680661603898, 2.77669639492884279630707952186, 4.11951947557934607660590485262, 4.96969117220031705662932151387, 5.79719877018855756154845499732, 6.76841303252218839365775602690, 7.10892159509430916408233832975, 7.70295918986399940301035943144, 8.826808267898864561329963552274
