| L(s) = 1 | + (89.6 − 155. i)3-s + (84.1 + 145. i)5-s + (3.16e3 + 5.50e3i)7-s + (−6.23e3 − 1.08e4i)9-s + (7.24e3 − 1.25e4i)11-s − 1.09e5·13-s + 3.01e4·15-s + (−1.50e5 + 2.60e5i)17-s + (−2.45e5 − 4.25e5i)19-s + (1.13e6 + 2.91e3i)21-s + (−1.13e6 − 1.96e6i)23-s + (9.62e5 − 1.66e6i)25-s + 1.29e6·27-s − 3.80e6·29-s + (3.95e6 − 6.84e6i)31-s + ⋯ |
| L(s) = 1 | + (0.639 − 1.10i)3-s + (0.0602 + 0.104i)5-s + (0.497 + 0.867i)7-s + (−0.316 − 0.548i)9-s + (0.149 − 0.258i)11-s − 1.05·13-s + 0.153·15-s + (−0.436 + 0.755i)17-s + (−0.432 − 0.749i)19-s + (1.27 + 0.00326i)21-s + (−0.845 − 1.46i)23-s + (0.492 − 0.853i)25-s + 0.468·27-s − 0.998·29-s + (0.768 − 1.33i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.556870920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.556870920\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-3.16e3 - 5.50e3i)T \) |
| good | 3 | \( 1 + (-89.6 + 155. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-84.1 - 145. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-7.24e3 + 1.25e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.09e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.50e5 - 2.60e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.45e5 + 4.25e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.13e6 + 1.96e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 3.80e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-3.95e6 + 6.84e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (7.07e6 + 1.22e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.11e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.50e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (2.18e7 + 3.78e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.75e7 + 4.77e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (2.45e7 - 4.24e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.09e7 - 1.22e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (4.93e7 - 8.55e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.96e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (2.30e7 - 3.98e7i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.76e8 + 4.78e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 5.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.54e8 + 4.40e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.18e9T + 7.60e17T^{2} \) |
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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
−11.68602904484511233345314366986, −10.34145611033947204032233694116, −8.859587567554194109107967219368, −8.200212292765399285975080025470, −7.07788027965738477234356019322, −5.99742754367875475437631270532, −4.46736953980933765300767257173, −2.52850641397383823977102304332, −2.01798333466545521781524354016, −0.33376412260019702287417379210,
1.54438664116810816854612677130, 3.16914262530171035108425509170, 4.25920057672168028208456698181, 5.13329727297004489738999334302, 7.00719881964466061227342037385, 8.073970216446253826980417583742, 9.363995144282521494869575562515, 9.981683622558661992984043825608, 11.01194127465757416240693929375, 12.20831353748863033462786986368
