L(s) = 1 | + 9.59i·2-s + (−8.65 + 12.9i)3-s − 60.0·4-s − 42.3·5-s + (−124. − 83.0i)6-s + 238. i·7-s − 269. i·8-s + (−93.1 − 224. i)9-s − 406. i·10-s + 261.·11-s + (519. − 778. i)12-s − 36.2·13-s − 2.29e3·14-s + (366. − 548. i)15-s + 662.·16-s + 1.88e3·17-s + ⋯ |
L(s) = 1 | + 1.69i·2-s + (−0.555 + 0.831i)3-s − 1.87·4-s − 0.757·5-s + (−1.41 − 0.941i)6-s + 1.84i·7-s − 1.48i·8-s + (−0.383 − 0.923i)9-s − 1.28i·10-s + 0.651·11-s + (1.04 − 1.56i)12-s − 0.0594·13-s − 3.12·14-s + (0.420 − 0.629i)15-s + 0.646·16-s + 1.57·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.610914 - 0.278066i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.610914 - 0.278066i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (8.65 - 12.9i)T \) |
| 23 | \( 1 + (666. - 2.44e3i)T \) |
good | 2 | \( 1 - 9.59iT - 32T^{2} \) |
| 5 | \( 1 + 42.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 238. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 261.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 36.2T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.88e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.30e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 + 2.23e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.19e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.39e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 104. iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.04e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.22e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.96e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.31e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.48e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 3.49e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.16e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 4.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.61e4iT - 8.58e9T^{2} \) |
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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
−14.97572135366819290871108781245, −14.39818211507297361470788134427, −12.33956974564281260535577156679, −11.63543700270146604344015392360, −9.660545463767386945374119886751, −8.787531288712149304957294510419, −7.64440352657380745444959328109, −5.93700046501717232252050256817, −5.45752520983824422133977222220, −3.82985857043809944806834221220,
0.37237203189763091650321085181, 1.31522861588620541541587232203, 3.40824395572123513522601390336, 4.60386370232914255091336355898, 6.90788236748419907742820843593, 8.068285149987747937091137054087, 9.923158206695158653246763414648, 10.84467936826779633992157476269, 11.63783008117706871733315652128, 12.50283974690728581228964773566