| L(s) = 1 | + (3.23 − 2.35i)2-s + (4.94 − 15.2i)4-s + (33.7 − 46.4i)5-s + (−94.7 − 30.7i)7-s + (−19.7 − 60.8i)8-s − 229. i·10-s + (−286. + 281. i)11-s + (−333. − 458. i)13-s + (−379. + 123. i)14-s + (−207. − 150. i)16-s + (−248. − 180. i)17-s + (−795. + 258. i)19-s + (−540. − 743. i)20-s + (−265. + 1.58e3i)22-s + 425. i·23-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.604 − 0.831i)5-s + (−0.730 − 0.237i)7-s + (−0.109 − 0.336i)8-s − 0.726i·10-s + (−0.713 + 0.700i)11-s + (−0.547 − 0.752i)13-s + (−0.516 + 0.167i)14-s + (−0.202 − 0.146i)16-s + (−0.208 − 0.151i)17-s + (−0.505 + 0.164i)19-s + (−0.302 − 0.415i)20-s + (−0.117 + 0.697i)22-s + 0.167i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9629381947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9629381947\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (286. - 281. i)T \) |
| good | 5 | \( 1 + (-33.7 + 46.4i)T + (-965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (94.7 + 30.7i)T + (1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (333. + 458. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (248. + 180. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (795. - 258. i)T + (2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 - 425. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (460. - 1.41e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-833. + 605. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-957. + 2.94e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (2.29e3 + 7.07e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 9.23e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (2.32e4 - 7.54e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.85e4 + 2.54e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (3.33e4 + 1.08e4i)T + (5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.43e4 - 1.98e4i)T + (-2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 - 2.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-2.65e4 + 3.64e4i)T + (-5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.19e4 - 3.89e3i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-6.17e4 - 8.50e4i)T + (-9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.20e4 + 8.76e3i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 4.44e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.54e4 + 4.75e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.03316239414933960867011299216, −9.999262792194871467261042823705, −9.420280813351393017719722448464, −7.993049744281507619484017211399, −6.68152684420768914305534630055, −5.46900082379090970691018848727, −4.64695483282154265690708124567, −3.13153916956850232986956423990, −1.81345546816430982478640658142, −0.21725005567437111413805127567,
2.30266037006328238722808929438, 3.27285815785849620808026742087, 4.80956285939042074300817784642, 6.14578939030059882003890241123, 6.64582485586124294465796915531, 7.938648611383097538310186553783, 9.198209520944772985915106613046, 10.24465487797309492630199978732, 11.18116704906536810944620945728, 12.33295370769940650608849465918
