| L(s) = 1 | + (−0.346 − 1.06i)2-s + (−0.809 + 0.587i)3-s + (0.598 − 0.435i)4-s + (0.407 − 2.19i)5-s + (0.907 + 0.659i)6-s + 1.11·7-s + (−2.48 − 1.80i)8-s + (0.309 − 0.951i)9-s + (−2.48 + 0.327i)10-s + (1.13 + 3.49i)11-s + (−0.228 + 0.704i)12-s + (−1.25 + 3.85i)13-s + (−0.386 − 1.18i)14-s + (0.962 + 2.01i)15-s + (−0.609 + 1.87i)16-s + (−1.71 − 1.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.245 − 0.754i)2-s + (−0.467 + 0.339i)3-s + (0.299 − 0.217i)4-s + (0.182 − 0.983i)5-s + (0.370 + 0.269i)6-s + 0.420·7-s + (−0.879 − 0.639i)8-s + (0.103 − 0.317i)9-s + (−0.786 + 0.103i)10-s + (0.341 + 1.05i)11-s + (−0.0660 + 0.203i)12-s + (−0.347 + 1.06i)13-s + (−0.103 − 0.317i)14-s + (0.248 + 0.521i)15-s + (−0.152 + 0.468i)16-s + (−0.416 − 0.302i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.674073 - 0.484601i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.674073 - 0.484601i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.407 + 2.19i)T \) |
| good | 2 | \( 1 + (0.346 + 1.06i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 + (-1.13 - 3.49i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.25 - 3.85i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.71 + 1.24i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-3.28 - 2.38i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.90 - 5.87i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.82 + 1.32i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (8.13 + 5.90i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.27 + 7.01i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.30 - 7.10i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + (-2.53 + 1.83i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.83 + 2.06i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.03 - 6.27i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.81 + 8.65i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (2.12 + 1.54i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (0.534 - 0.388i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.31 - 7.10i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (6.90 - 5.01i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (9.92 + 7.20i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (4.72 + 14.5i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (10.9 - 7.93i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.39066387807726478705953168692, −12.84842181519783517605402778245, −11.88247434127702323943591370822, −11.24235897815236521917048905167, −9.704242955100650272378055763923, −9.283991757801920567827831263075, −7.27681945261482989481994729666, −5.66615835151269864645800604719, −4.27783528232799575203852884513, −1.73835891101938825072449857934,
2.93910431623978555694765773375, 5.53867550354446828730854890838, 6.61011354759931948323651842407, 7.55479382414135251951773911664, 8.743292264395658630734977098889, 10.62179614738553591884322119741, 11.31477110731725426850840535498, 12.52430681795716310452407045627, 13.96592975472681378418021740481, 14.87723787404062556803689378623
