| L(s) = 1 | + (0.978 + 0.207i)2-s + (−0.900 + 1.47i)3-s + (0.913 + 0.406i)4-s + (−0.0601 − 2.23i)5-s + (−1.18 + 1.26i)6-s + (1.88 − 3.27i)7-s + (0.809 + 0.587i)8-s + (−1.37 − 2.66i)9-s + (0.405 − 2.19i)10-s + (3.32 + 0.706i)11-s + (−1.42 + 0.985i)12-s + (0.204 − 0.0435i)13-s + (2.52 − 2.80i)14-s + (3.36 + 1.92i)15-s + (0.669 + 0.743i)16-s + (1.78 + 1.30i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (−0.519 + 0.854i)3-s + (0.456 + 0.203i)4-s + (−0.0268 − 0.999i)5-s + (−0.485 + 0.514i)6-s + (0.713 − 1.23i)7-s + (0.286 + 0.207i)8-s + (−0.459 − 0.887i)9-s + (0.128 − 0.695i)10-s + (1.00 + 0.213i)11-s + (−0.411 + 0.284i)12-s + (0.0567 − 0.0120i)13-s + (0.675 − 0.750i)14-s + (0.868 + 0.496i)15-s + (0.167 + 0.185i)16-s + (0.434 + 0.315i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.92780 - 0.0999039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92780 - 0.0999039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.900 - 1.47i)T \) |
| 5 | \( 1 + (0.0601 + 2.23i)T \) |
| good | 7 | \( 1 + (-1.88 + 3.27i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.32 - 0.706i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.204 + 0.0435i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.78 - 1.30i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (4.15 + 3.02i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.91 + 3.23i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.958 - 9.11i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.345 - 3.29i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (1.69 - 5.20i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-9.19 + 1.95i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (0.588 - 1.01i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.547 + 5.20i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.32 + 0.965i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (9.79 - 2.08i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (9.03 + 1.92i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.11 + 10.6i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (9.77 - 7.09i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.63 + 8.12i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.347 - 3.30i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.95 + 0.872i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.29 - 10.1i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.820 - 7.80i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−10.98144484093128824976499852631, −10.52363565209277472324544059359, −9.244444620803847531206742012832, −8.473305910225188980014655224623, −7.17947817964237353506964865813, −6.22533191752647567553272888663, −4.90208461358462890595441467314, −4.52254402518324602473197594032, −3.57296046338749078836995843011, −1.22558985291407043726354669542,
1.78888674945862312604045191748, 2.80459381645821958957018120941, 4.32216693734782901345349828578, 5.85043947370067367761203887704, 6.03667246311035433709485750466, 7.26141857421670386121945524127, 8.097593089392189140056893191213, 9.335291239369672989242761433957, 10.65906345787754880061121341543, 11.46859117257975254055524641274
