L(s) = 1 | − 2.51e3i·2-s + (−4.46e5 − 2.88e5i)3-s + 1.04e7·4-s − 3.68e8i·5-s + (−7.25e8 + 1.12e9i)6-s − 8.30e9·7-s − 6.84e10i·8-s + (1.15e11 + 2.57e11i)9-s − 9.24e11·10-s + 3.87e12i·11-s + (−4.67e12 − 3.02e12i)12-s − 3.48e13·13-s + 2.08e13i·14-s + (−1.06e14 + 1.64e14i)15-s + 3.82e12·16-s − 2.61e14i·17-s + ⋯ |
L(s) = 1 | − 0.613i·2-s + (−0.839 − 0.543i)3-s + 0.624·4-s − 1.50i·5-s + (−0.333 + 0.514i)6-s − 0.600·7-s − 0.995i·8-s + (0.409 + 0.912i)9-s − 0.924·10-s + 1.23i·11-s + (−0.523 − 0.339i)12-s − 1.49·13-s + 0.368i·14-s + (−0.819 + 1.26i)15-s + 0.0135·16-s − 0.449i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{25}{2})\) |
\(\approx\) |
\(0.235552 + 0.797239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.235552 + 0.797239i\) |
\(L(13)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.46e5 + 2.88e5i)T \) |
good | 2 | \( 1 + 2.51e3iT - 1.67e7T^{2} \) |
| 5 | \( 1 + 3.68e8iT - 5.96e16T^{2} \) |
| 7 | \( 1 + 8.30e9T + 1.91e20T^{2} \) |
| 11 | \( 1 - 3.87e12iT - 9.84e24T^{2} \) |
| 13 | \( 1 + 3.48e13T + 5.42e26T^{2} \) |
| 17 | \( 1 + 2.61e14iT - 3.39e29T^{2} \) |
| 19 | \( 1 - 7.24e14T + 4.89e30T^{2} \) |
| 23 | \( 1 + 1.12e16iT - 4.80e32T^{2} \) |
| 29 | \( 1 - 1.35e17iT - 1.25e35T^{2} \) |
| 31 | \( 1 - 2.21e17T + 6.20e35T^{2} \) |
| 37 | \( 1 + 1.03e19T + 4.33e37T^{2} \) |
| 41 | \( 1 + 1.85e19iT - 5.09e38T^{2} \) |
| 43 | \( 1 - 2.09e19T + 1.59e39T^{2} \) |
| 47 | \( 1 + 5.93e19iT - 1.35e40T^{2} \) |
| 53 | \( 1 + 1.46e20iT - 2.41e41T^{2} \) |
| 59 | \( 1 + 3.04e21iT - 3.16e42T^{2} \) |
| 61 | \( 1 - 3.55e21T + 7.04e42T^{2} \) |
| 67 | \( 1 + 2.84e21T + 6.69e43T^{2} \) |
| 71 | \( 1 + 2.39e22iT - 2.69e44T^{2} \) |
| 73 | \( 1 + 1.66e22T + 5.24e44T^{2} \) |
| 79 | \( 1 + 6.08e22T + 3.49e45T^{2} \) |
| 83 | \( 1 - 7.63e22iT - 1.14e46T^{2} \) |
| 89 | \( 1 - 1.24e23iT - 6.10e46T^{2} \) |
| 97 | \( 1 - 9.42e23T + 4.81e47T^{2} \) |
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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
−19.44635431756050667261985337768, −17.25295209562531826941242748180, −16.02037027806004813820156377549, −12.68697853670343120713327880852, −12.10788183534108869917706994055, −9.890131449050827775582157960062, −7.11665161871161613171616554627, −4.94145017193832196632629452459, −1.93949047850805886881082440041, −0.39189162539001633464885667364,
3.02855356145617913127565593212, 5.87481593353681542930751611414, 7.07245650441216570917958217340, 10.29037774195527955284593870794, 11.58466459425104290781776732412, 14.60680643996933495508973406653, 15.88949061805608343862059392396, 17.32224286605742360039748284801, 19.22235875365080858445507652969, 21.61445227155807054671955074445