| L(s) = 1 | + (5.65 − 5.65i)2-s + (−16.7 + 43.6i)3-s − 64.0i·4-s + (151. + 341. i)6-s + (469. + 469. i)7-s + (−362. − 362. i)8-s + (−1.62e3 − 1.46e3i)9-s − 2.97e3i·11-s + (2.79e3 + 1.07e3i)12-s + (−5.60e3 + 5.60e3i)13-s + 5.31e3·14-s − 4.09e3·16-s + (−7.32e3 + 7.32e3i)17-s + (−1.74e4 + 891. i)18-s − 4.78e4i·19-s + ⋯ |
| L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.359 + 0.933i)3-s − 0.500i·4-s + (0.287 + 0.646i)6-s + (0.517 + 0.517i)7-s + (−0.250 − 0.250i)8-s + (−0.742 − 0.670i)9-s − 0.674i·11-s + (0.466 + 0.179i)12-s + (−0.708 + 0.708i)13-s + 0.517·14-s − 0.250·16-s + (−0.361 + 0.361i)17-s + (−0.706 + 0.0360i)18-s − 1.60i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.459238248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.459238248\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-5.65 + 5.65i)T \) |
| 3 | \( 1 + (16.7 - 43.6i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-469. - 469. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + 2.97e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + (5.60e3 - 5.60e3i)T - 6.27e7iT^{2} \) |
| 17 | \( 1 + (7.32e3 - 7.32e3i)T - 4.10e8iT^{2} \) |
| 19 | \( 1 + 4.78e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + (-4.77e4 - 4.77e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + 7.10e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (4.20e5 + 4.20e5i)T + 9.49e10iT^{2} \) |
| 41 | \( 1 + 1.82e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (-2.70e5 + 2.70e5i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (-4.82e5 + 4.82e5i)T - 5.06e11iT^{2} \) |
| 53 | \( 1 + (1.21e6 + 1.21e6i)T + 1.17e12iT^{2} \) |
| 59 | \( 1 - 1.15e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.49e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + (6.72e5 + 6.72e5i)T + 6.06e12iT^{2} \) |
| 71 | \( 1 + 4.36e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (-3.38e6 + 3.38e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 - 5.85e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (5.25e6 + 5.25e6i)T + 2.71e13iT^{2} \) |
| 89 | \( 1 + 9.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-9.89e6 - 9.89e6i)T + 8.07e13iT^{2} \) |
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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
−11.35606132048077852572123093019, −10.71731036774893848292256956853, −9.404673808119722748177660549541, −8.742288572984394324659434955839, −6.88445244927677627984762494731, −5.51218899405586522350326808377, −4.78034171666662849307681871397, −3.54969201763660988080819037405, −2.22089619110177217683046629324, −0.36006944503673612869477264624,
1.25403707936819888034976553888, 2.71607545236771331579096005254, 4.48006416713219160187973083848, 5.50053112024621358463789808786, 6.72046170921574969965721044008, 7.56268121465374817169039589900, 8.361047608911699085174989722880, 10.06501479679019981497971860606, 11.19860744686038034953103770701, 12.29922488709538304441136058323
