| L(s) = 1 | + (−0.395 + 0.776i)2-s + (−0.296 + 1.87i)3-s + (1.90 + 2.62i)4-s + (1.22 − 4.84i)5-s + (−1.33 − 0.970i)6-s + (−5.60 − 5.60i)7-s + (−6.23 + 0.986i)8-s + (5.14 + 1.67i)9-s + (3.27 + 2.87i)10-s + (−4.46 − 13.7i)11-s + (−5.47 + 2.78i)12-s + (5.52 + 10.8i)13-s + (6.57 − 2.13i)14-s + (8.70 + 3.73i)15-s + (−2.30 + 7.10i)16-s + (0.147 + 0.932i)17-s + ⋯ |
| L(s) = 1 | + (−0.197 + 0.388i)2-s + (−0.0988 + 0.623i)3-s + (0.476 + 0.655i)4-s + (0.245 − 0.969i)5-s + (−0.222 − 0.161i)6-s + (−0.801 − 0.801i)7-s + (−0.778 + 0.123i)8-s + (0.571 + 0.185i)9-s + (0.327 + 0.287i)10-s + (−0.406 − 1.25i)11-s + (−0.456 + 0.232i)12-s + (0.425 + 0.834i)13-s + (0.469 − 0.152i)14-s + (0.580 + 0.249i)15-s + (−0.144 + 0.443i)16-s + (0.00869 + 0.0548i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.808417 + 0.352644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.808417 + 0.352644i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.22 + 4.84i)T \) |
| good | 2 | \( 1 + (0.395 - 0.776i)T + (-2.35 - 3.23i)T^{2} \) |
| 3 | \( 1 + (0.296 - 1.87i)T + (-8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (5.60 + 5.60i)T + 49iT^{2} \) |
| 11 | \( 1 + (4.46 + 13.7i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-5.52 - 10.8i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-0.147 - 0.932i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (11.5 - 15.8i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (4.69 + 2.39i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (7.87 + 10.8i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-12.1 - 8.83i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-57.7 + 29.4i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (16.7 - 51.6i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-3.47 + 3.47i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-65.5 - 10.3i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (6.61 - 41.7i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (35.6 + 11.5i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (32.4 + 99.9i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (7.05 + 44.5i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-36.5 + 26.5i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-47.8 - 24.3i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-70.5 - 97.1i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (60.5 - 9.59i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-48.8 + 15.8i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (36.3 + 5.75i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−16.79398966767949417503821063580, −16.53999584329451062955091405039, −15.69295874375335105614944072169, −13.64883627917978882192818148673, −12.59350450130060521807649699863, −10.93357521905109734423185663042, −9.438343075774844698712396795800, −8.024756812985741207509298017224, −6.22477370456599607984679984206, −3.97061759697752838619753611061,
2.46728936999697706176841423502, 6.04647232390251100700420531164, 7.14573973507758729723259673492, 9.590891891808325520972965546862, 10.57066273174379809289595777296, 12.09197200985046640482649261615, 13.19522294035880671376233948886, 15.09628056458961582926411190645, 15.53696694371917354545787734367, 17.82396037574020397660836976814
