| L(s) = 1 | + (0.655 + 2.01i)2-s + (−0.809 + 0.587i)3-s + (−2.02 + 1.47i)4-s + (−2.21 − 0.291i)5-s + (−1.71 − 1.24i)6-s + 4.35·7-s + (−0.865 − 0.629i)8-s + (0.309 − 0.951i)9-s + (−0.865 − 4.66i)10-s + (−0.488 − 1.50i)11-s + (0.773 − 2.38i)12-s + (0.370 − 1.13i)13-s + (2.85 + 8.79i)14-s + (1.96 − 1.06i)15-s + (−0.845 + 2.60i)16-s + (0.907 + 0.659i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 1.42i)2-s + (−0.467 + 0.339i)3-s + (−1.01 + 0.735i)4-s + (−0.991 − 0.130i)5-s + (−0.700 − 0.509i)6-s + 1.64·7-s + (−0.306 − 0.222i)8-s + (0.103 − 0.317i)9-s + (−0.273 − 1.47i)10-s + (−0.147 − 0.453i)11-s + (0.223 − 0.687i)12-s + (0.102 − 0.315i)13-s + (0.763 + 2.35i)14-s + (0.507 − 0.275i)15-s + (−0.211 + 0.650i)16-s + (0.220 + 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.535167 + 0.834022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.535167 + 0.834022i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.21 + 0.291i)T \) |
| good | 2 | \( 1 + (-0.655 - 2.01i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 11 | \( 1 + (0.488 + 1.50i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.370 + 1.13i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.907 - 0.659i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (6.21 + 4.51i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.717 + 2.20i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.45 + 3.23i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.88 + 2.82i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.96 - 6.06i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.30 - 7.10i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 + (3.33 - 2.42i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.03 - 2.20i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.82 + 8.70i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.431 - 1.32i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.12 - 2.27i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-8.57 + 6.23i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.54 - 4.75i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (11.7 - 8.55i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.06 - 5.13i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (3.10 + 9.54i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.06 + 4.40i)T + (29.9 - 92.2i)T^{2} \) |
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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
−15.07860776283231807152491407309, −14.32127835822386016303443565505, −12.96067588103180392866593340744, −11.52638759707711784142060678005, −10.80268730825703865331451743131, −8.487268697606685298308983033273, −7.968672936935089212692455921243, −6.56765217069018723970081181290, −5.09091111200148148826686848611, −4.33030130136225072478253153961,
1.81778738923904851338622341369, 3.97878051966837113801854979187, 5.04760304503068009493116287523, 7.28187870017750320488905422969, 8.473063663934569334960750903380, 10.46584642318804425484130187317, 11.10841447277144338927229419482, 11.99080141989912704156449019209, 12.57369625207044956796655481063, 14.05740719790191870490265412022
