| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−12.9 + 9.40i)4-s + (7.80 − 24.0i)5-s + (84.3 − 61.2i)7-s + (51.7 + 37.6i)8-s − 101.·10-s + (301. + 265. i)11-s + (144. + 445. i)13-s + (−337. − 245. i)14-s + (79.1 − 243. i)16-s + (−524. + 1.61e3i)17-s + (−1.73e3 − 1.26e3i)19-s + (124. + 384. i)20-s + (637. − 1.47e3i)22-s + 2.08e3·23-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.139 − 0.429i)5-s + (0.650 − 0.472i)7-s + (0.286 + 0.207i)8-s − 0.319·10-s + (0.750 + 0.661i)11-s + (0.237 + 0.730i)13-s + (−0.460 − 0.334i)14-s + (0.0772 − 0.237i)16-s + (−0.439 + 1.35i)17-s + (−1.10 − 0.802i)19-s + (0.0698 + 0.214i)20-s + (0.280 − 0.649i)22-s + 0.820·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.904567715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904567715\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-301. - 265. i)T \) |
| good | 5 | \( 1 + (-7.80 + 24.0i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-84.3 + 61.2i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-144. - 445. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (524. - 1.61e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (1.73e3 + 1.26e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 2.08e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (4.30e3 - 3.12e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-193. - 596. i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.32e4 + 9.60e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.56e4 - 1.13e4i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.71e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.40e3 + 1.74e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.69e3 + 8.30e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (6.64e3 - 4.82e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.58e4 + 4.89e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 6.44e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (7.85e3 - 2.41e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.04e4 + 2.93e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-4.82e3 - 1.48e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.78e4 + 8.56e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + 523.T + 5.58e9T^{2} \) |
| 97 | \( 1 + (4.43e3 + 1.36e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−11.14053619809163343386454088718, −10.93822583294592832245866970458, −9.388071522029128004683735323344, −8.850226080048941628205739864781, −7.60017641514818628428951995169, −6.38521366492143647869764483931, −4.70782396315486860030109517919, −3.98458132232939485783736968499, −2.09979570037750810806343553445, −1.07120788960829802199265997064,
0.806331903990508432690554065032, 2.58489989101509568104429510798, 4.24620957990614411078202350180, 5.57173634235892677341812207007, 6.44718326981739868919359667741, 7.63165586862120670870675327320, 8.603102964705938355805278570695, 9.446172251796204304114756983010, 10.74364851582025739761807158726, 11.47046307118000120012379853291
