| L(s) = 1 | + (−0.358 − 0.984i)2-s + (0.523 + 0.0922i)3-s + (0.691 − 0.580i)4-s + (0.296 + 2.21i)5-s + (−0.0966 − 0.548i)6-s + (2.37 − 1.37i)7-s + (−2.63 − 1.52i)8-s + (−2.55 − 0.929i)9-s + (2.07 − 1.08i)10-s + (−0.416 + 0.721i)11-s + (0.415 − 0.239i)12-s + (−0.601 + 0.106i)13-s + (−2.19 − 1.84i)14-s + (−0.0492 + 1.18i)15-s + (−0.239 + 1.35i)16-s + (1.65 + 4.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 − 0.695i)2-s + (0.302 + 0.0532i)3-s + (0.345 − 0.290i)4-s + (0.132 + 0.991i)5-s + (−0.0394 − 0.223i)6-s + (0.896 − 0.517i)7-s + (−0.930 − 0.537i)8-s + (−0.851 − 0.309i)9-s + (0.656 − 0.343i)10-s + (−0.125 + 0.217i)11-s + (0.119 − 0.0692i)12-s + (−0.166 + 0.0294i)13-s + (−0.587 − 0.493i)14-s + (−0.0127 + 0.306i)15-s + (−0.0598 + 0.339i)16-s + (0.401 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.960666 - 0.426068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.960666 - 0.426068i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.296 - 2.21i)T \) |
| 19 | \( 1 + (4.35 - 0.175i)T \) |
| good | 2 | \( 1 + (0.358 + 0.984i)T + (-1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (-0.523 - 0.0922i)T + (2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (-2.37 + 1.37i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.416 - 0.721i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.601 - 0.106i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.65 - 4.54i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.41 - 2.87i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (3.73 + 1.35i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.46 - 5.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.33iT - 37T^{2} \) |
| 41 | \( 1 + (-0.923 + 5.23i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.72 + 8.01i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (1.16 - 3.19i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (8.78 + 10.4i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-9.41 + 3.42i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.94 + 5.83i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (3.73 - 10.2i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (0.519 + 0.435i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-6.90 - 1.21i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.604 - 3.42i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (4.30 - 2.48i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.02 - 5.79i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (4.25 + 11.6i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−14.19219336747174246664317693231, −12.58524195446852306253212735440, −11.31775326826667314562870566577, −10.81013714723730878073875576680, −9.825071347044405987879138029531, −8.427972595996825468375648118642, −7.05908358382443831775277743596, −5.78453122463407322918519604673, −3.60357291254788385128034995701, −2.11164753799726985701026636410,
2.53947229268890639505226491747, 4.91401413297507276310351715115, 6.02677241060336968219148791863, 7.74681340276455305615347408180, 8.411580990748362592172975090096, 9.259719637221316872497749546226, 11.20158960882565352947505978137, 11.95501967162237824287513356762, 13.13380365373940144687385798857, 14.39924422255063853474249908860
