| L(s) = 1 | + (33.0 + 33.0i)3-s + (127. − 127. i)7-s + 2.18e3i·9-s − 3.46e3·11-s + (9.01e3 + 9.01e3i)13-s + (5.07e4 − 5.07e4i)17-s − 2.38e4i·19-s + 8.46e3·21-s + (−3.02e5 − 3.02e5i)23-s + (−7.23e4 + 7.23e4i)27-s + 1.01e6i·29-s + 5.19e5·31-s + (−1.14e5 − 1.14e5i)33-s + (1.86e6 − 1.86e6i)37-s + 5.96e5i·39-s + ⋯ |
| L(s) = 1 | + (0.408 + 0.408i)3-s + (0.0532 − 0.0532i)7-s + 0.333i·9-s − 0.236·11-s + (0.315 + 0.315i)13-s + (0.607 − 0.607i)17-s − 0.182i·19-s + 0.0435·21-s + (−1.08 − 1.08i)23-s + (−0.136 + 0.136i)27-s + 1.43i·29-s + 0.562·31-s + (−0.0965 − 0.0965i)33-s + (0.997 − 0.997i)37-s + 0.257i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.444962918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.444962918\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-33.0 - 33.0i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-127. + 127. i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + 3.46e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-9.01e3 - 9.01e3i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 + (-5.07e4 + 5.07e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + 2.38e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (3.02e5 + 3.02e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 - 1.01e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 5.19e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (-1.86e6 + 1.86e6i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 - 1.25e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (2.86e6 + 2.86e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + (-8.88e5 + 8.88e5i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 + (3.11e6 + 3.11e6i)T + 6.22e13iT^{2} \) |
| 59 | \( 1 + 9.36e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.53e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + (-9.05e6 + 9.05e6i)T - 4.06e14iT^{2} \) |
| 71 | \( 1 - 7.00e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.60e7 - 1.60e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 1.60e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-2.76e7 - 2.76e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 + 8.39e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-8.71e7 + 8.71e7i)T - 7.83e15iT^{2} \) |
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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
−10.26449670302158582290103816117, −9.372717713116665642846645475389, −8.471869298527691475694333064842, −7.56678189720041317944951904374, −6.42953939672209818195360378640, −5.23641710240033336495508932493, −4.22886623144884860366206729693, −3.14040654655656370807472449260, −2.02488795119526871911360079827, −0.59207104201329062643715940255,
0.858570638317533064212278560599, 1.96305132701535086649464093404, 3.13981727918943450910229770272, 4.20138975401848837713377940160, 5.62347260118512624822531244648, 6.44506985239487846506737755834, 7.83668268111909500103863290519, 8.158744653772880028157003528129, 9.511575150881635926171931348589, 10.21496122595192474514805232041
