| L(s) = 1 | + (−0.109 + 0.0793i)2-s + (−2.46 + 7.59i)4-s + (6.22 − 9.28i)5-s − 17.3·7-s + (−0.666 − 2.05i)8-s + (0.0573 + 1.50i)10-s + (34.1 − 24.7i)11-s + (68.3 + 49.6i)13-s + (1.89 − 1.37i)14-s + (−51.4 − 37.3i)16-s + (13.9 + 42.8i)17-s + (25.8 + 79.5i)19-s + (55.1 + 70.1i)20-s + (−1.75 + 5.41i)22-s + (16.9 − 12.3i)23-s + ⋯ |
| L(s) = 1 | + (−0.0386 + 0.0280i)2-s + (−0.308 + 0.948i)4-s + (0.556 − 0.830i)5-s − 0.934·7-s + (−0.0294 − 0.0906i)8-s + (0.00181 + 0.0476i)10-s + (0.934 − 0.679i)11-s + (1.45 + 1.05i)13-s + (0.0360 − 0.0262i)14-s + (−0.803 − 0.583i)16-s + (0.198 + 0.611i)17-s + (0.312 + 0.960i)19-s + (0.616 + 0.784i)20-s + (−0.0170 + 0.0524i)22-s + (0.153 − 0.111i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.737i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.674 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.55134 + 0.683490i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55134 + 0.683490i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-6.22 + 9.28i)T \) |
| good | 2 | \( 1 + (0.109 - 0.0793i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 + 17.3T + 343T^{2} \) |
| 11 | \( 1 + (-34.1 + 24.7i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-68.3 - 49.6i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-13.9 - 42.8i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-25.8 - 79.5i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-16.9 + 12.3i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (68.0 - 209. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-49.5 - 152. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-212. - 154. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-331. - 241. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-142. + 438. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-44.0 + 135. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (382. + 277. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (43.7 - 31.7i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-139. - 427. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-193. + 595. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (39.4 - 28.6i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (200. - 615. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (206. + 637. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (700. - 509. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (54.7 - 168. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.15671370857237155904796308245, −11.09666638834958101815815391741, −9.637852537651459743852757167103, −8.934411074901456073300237444543, −8.283736057942333214791529551630, −6.70705311834590479510239007362, −5.88660902773021118806626873653, −4.21151830071040823933350590906, −3.36020675777608409590039025157, −1.29632242523553436299755543699,
0.861828291890393950605054734939, 2.62272324455015365400409580174, 4.07879628006360119977916375979, 5.79722973136111046050003459948, 6.26673668809570775255726215414, 7.45736063881975691253346143640, 9.275447509109960793122219570070, 9.561213150637066437797284826229, 10.66732620993889436228915235002, 11.34889011977846163149781584423
