| L(s) = 1 | + (−179. + 479. i)2-s + 1.13e4i·3-s + (−1.97e5 − 1.72e5i)4-s + 1.99e6·5-s + (−5.44e6 − 2.03e6i)6-s + 1.75e7i·7-s + (1.17e8 − 6.39e7i)8-s − 1.29e8·9-s + (−3.57e8 + 9.56e8i)10-s + 2.97e9i·11-s + (1.95e9 − 2.24e9i)12-s − 1.95e10·13-s + (−8.41e9 − 3.14e9i)14-s + 2.26e10i·15-s + (9.50e9 + 6.80e10i)16-s + 8.74e10·17-s + ⋯ |
| L(s) = 1 | + (−0.350 + 0.936i)2-s + 0.577i·3-s + (−0.754 − 0.656i)4-s + 1.02·5-s + (−0.540 − 0.202i)6-s + 0.434i·7-s + (0.879 − 0.476i)8-s − 0.333·9-s + (−0.357 + 0.956i)10-s + 1.26i·11-s + (0.378 − 0.435i)12-s − 1.84·13-s + (−0.407 − 0.152i)14-s + 0.589i·15-s + (0.138 + 0.990i)16-s + 0.737·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.7534342008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7534342008\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (179. - 479. i)T \) |
| 3 | \( 1 - 1.13e4iT \) |
| good | 5 | \( 1 - 1.99e6T + 3.81e12T^{2} \) |
| 7 | \( 1 - 1.75e7iT - 1.62e15T^{2} \) |
| 11 | \( 1 - 2.97e9iT - 5.55e18T^{2} \) |
| 13 | \( 1 + 1.95e10T + 1.12e20T^{2} \) |
| 17 | \( 1 - 8.74e10T + 1.40e22T^{2} \) |
| 19 | \( 1 - 2.99e11iT - 1.04e23T^{2} \) |
| 23 | \( 1 + 1.60e12iT - 3.24e24T^{2} \) |
| 29 | \( 1 + 1.54e13T + 2.10e26T^{2} \) |
| 31 | \( 1 + 3.94e13iT - 6.99e26T^{2} \) |
| 37 | \( 1 + 6.20e13T + 1.68e28T^{2} \) |
| 41 | \( 1 + 9.34e13T + 1.07e29T^{2} \) |
| 43 | \( 1 - 3.53e14iT - 2.52e29T^{2} \) |
| 47 | \( 1 - 1.01e15iT - 1.25e30T^{2} \) |
| 53 | \( 1 + 4.16e15T + 1.08e31T^{2} \) |
| 59 | \( 1 + 4.45e15iT - 7.50e31T^{2} \) |
| 61 | \( 1 + 1.18e16T + 1.36e32T^{2} \) |
| 67 | \( 1 + 1.84e16iT - 7.40e32T^{2} \) |
| 71 | \( 1 - 5.36e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 8.95e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + 1.71e15iT - 1.43e34T^{2} \) |
| 83 | \( 1 - 1.51e17iT - 3.49e34T^{2} \) |
| 89 | \( 1 - 1.49e17T + 1.22e35T^{2} \) |
| 97 | \( 1 - 1.47e18T + 5.77e35T^{2} \) |
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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
−16.75026490210951536088098684194, −15.07772261006834424482958890712, −14.36102941610177308997169537102, −12.54897319267178771855207423794, −10.01983245154762244450053084222, −9.506195029801553286224545975534, −7.57886320381097975945842070224, −5.86417371090806789417835698862, −4.68539329564648092634057239249, −2.04617399489018539415472459087,
0.27143425318239200674859082589, 1.69406932105152598435761736602, 3.06166010985305937588410963838, 5.31404990178312837164390756457, 7.40188000766922068185482809690, 9.128241896381739557018500217306, 10.39058360274295530324290736976, 11.90714300697448998953545398645, 13.29195857298002723244895965730, 14.16505748097068346170876024784
