| L(s) = 1 | − 5.00·2-s + 17.0·4-s + 7.07i·5-s − 10.4i·7-s − 45.5·8-s − 35.4i·10-s + (−13.0 − 34.0i)11-s + 43.7i·13-s + 52.4i·14-s + 91.2·16-s + 115.·17-s + 89.3i·19-s + 120. i·20-s + (65.4 + 170. i)22-s + 213. i·23-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.13·4-s + 0.632i·5-s − 0.565i·7-s − 2.01·8-s − 1.11i·10-s + (−0.358 − 0.933i)11-s + 0.932i·13-s + 1.00i·14-s + 1.42·16-s + 1.64·17-s + 1.07i·19-s + 1.35i·20-s + (0.634 + 1.65i)22-s + 1.93i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.553058 + 0.295597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.553058 + 0.295597i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (13.0 + 34.0i)T \) |
| good | 2 | \( 1 + 5.00T + 8T^{2} \) |
| 5 | \( 1 - 7.07iT - 125T^{2} \) |
| 7 | \( 1 + 10.4iT - 343T^{2} \) |
| 13 | \( 1 - 43.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 89.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 213. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 149.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 172.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 258. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 240. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 334. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 382. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 609. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 260T + 3.00e5T^{2} \) |
| 71 | \( 1 + 17.3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 787. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 11.1iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.79e3T + 9.12e5T^{2} \) |
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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
−13.75977322543258720876861977556, −11.97402586660741445439961276172, −11.03783579164371336922408025704, −10.22219742946359661512166214900, −9.324905199884385657068769338272, −8.007686385364032265363036912818, −7.28978805332794766963914698763, −5.97724073314869673571793506554, −3.26974492363847643678859257167, −1.30008869453229184012365063684,
0.72795141777962244672796958517, 2.52547317147681956120262272061, 5.20967757345229288175472990744, 6.86692278841092766955639420509, 8.038828585805256917547876735366, 8.804868506896879132669467353918, 9.893256841131316145379816971584, 10.64647298922751666724973175054, 12.05806569056965102567747483584, 12.77575246115107922383995746374
