| L(s) = 1 | + (35.0 + 73.0i)3-s − 4.17e3·7-s + (−4.10e3 + 5.11e3i)9-s + 8.71e3i·11-s − 1.63e4·13-s + 8.95e4i·17-s − 1.65e5·19-s + (−1.46e5 − 3.05e5i)21-s + 4.46e5i·23-s + (−5.17e5 − 1.20e5i)27-s + 1.40e6i·29-s + 8.20e5·31-s + (−6.36e5 + 3.05e5i)33-s − 2.19e5·37-s + (−5.74e5 − 1.19e6i)39-s + ⋯ |
| L(s) = 1 | + (0.432 + 0.901i)3-s − 1.74·7-s + (−0.625 + 0.779i)9-s + 0.595i·11-s − 0.574·13-s + 1.07i·17-s − 1.27·19-s + (−0.752 − 1.56i)21-s + 1.59i·23-s + (−0.973 − 0.226i)27-s + 1.98i·29-s + 0.887·31-s + (−0.536 + 0.257i)33-s − 0.117·37-s + (−0.248 − 0.517i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.1408374007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1408374007\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-35.0 - 73.0i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.17e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 8.71e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.63e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 8.95e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.65e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.46e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.40e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 8.20e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.19e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.78e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.47e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.99e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 9.07e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 9.44e5iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 7.30e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.58e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.16e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.08e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 6.47e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.12e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.43e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.29e8T + 7.83e15T^{2} \) |
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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
−10.70191222741522978927131503839, −10.10619477109549403330073045631, −9.364913628421556878265230185452, −8.559304430907591549480563368167, −7.24228709727210030897826231606, −6.26052298817231934822995933982, −5.11566002628483559262321713814, −3.87538948919651445798268982803, −3.19218439941378898263223271913, −1.97196665773014739657197072128,
0.03879981790019337183862862323, 0.66425291116710845582678824431, 2.46371926196683697387227384579, 2.97370102803661177608632614294, 4.34653003159402871282690543196, 6.16642197940923060506825195436, 6.46085224631077605843627646049, 7.58656520132490994376980885825, 8.631533237437179415508634258558, 9.474481045717671696695652197170
