| L(s) = 1 | + (−15.3 − 4.64i)2-s + 75.7·3-s + (212. + 142. i)4-s + (−1.15e3 − 351. i)6-s − 210.·7-s + (−2.60e3 − 3.16e3i)8-s − 823.·9-s − 141. i·11-s + (1.61e4 + 1.07e4i)12-s − 1.76e4i·13-s + (3.22e3 + 976. i)14-s + (2.51e4 + 6.05e4i)16-s + 1.02e5i·17-s + (1.26e4 + 3.81e3i)18-s + 1.07e5i·19-s + ⋯ |
| L(s) = 1 | + (−0.957 − 0.290i)2-s + 0.935·3-s + (0.831 + 0.555i)4-s + (−0.894 − 0.271i)6-s − 0.0876·7-s + (−0.635 − 0.772i)8-s − 0.125·9-s − 0.00967i·11-s + (0.777 + 0.519i)12-s − 0.618i·13-s + (0.0838 + 0.0254i)14-s + (0.383 + 0.923i)16-s + 1.22i·17-s + (0.120 + 0.0363i)18-s + 0.821i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.881413 + 0.777726i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.881413 + 0.777726i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (15.3 + 4.64i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 75.7T + 6.56e3T^{2} \) |
| 7 | \( 1 + 210.T + 5.76e6T^{2} \) |
| 11 | \( 1 + 141. iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.76e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.02e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.07e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 4.55e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 8.65e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 4.29e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.51e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.98e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 2.22e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 7.63e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 1.55e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 6.38e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.03e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.03e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 4.44e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.79e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.03e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 5.09e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 2.68e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 3.38e7iT - 7.83e15T^{2} \) |
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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
−12.48267397218804964469954422237, −11.19455724227666121987502312862, −10.21800243059818212238812154268, −9.109729235312659012131093579472, −8.322189081195524003054897254138, −7.40833362088262785064669368832, −5.90811515126426938808532728357, −3.68912084069493367306084390960, −2.66886630245090583057301651618, −1.32181635623518679226456350833,
0.40351504589852197422590029408, 2.05284333765794749616276527082, 3.17239155099017873212271105313, 5.20467709270118733390453973277, 6.81144567283220552432352777877, 7.67647253515749013084825903623, 9.092218687146526313567162393236, 9.219587516109296489450923709647, 10.82830691440973519001128833701, 11.71372146629127587461260492220
