L(s) = 1 | + 2.89e3i·2-s + (−3.17e5 − 4.25e5i)3-s − 8.38e6·4-s + 6.76e7i·5-s + (1.23e9 − 9.20e8i)6-s − 2.41e10·7-s − 2.42e10i·8-s + (−8.02e10 + 2.70e11i)9-s − 1.96e11·10-s − 4.00e12i·11-s + (2.66e12 + 3.57e12i)12-s + 2.67e13·13-s − 6.99e13i·14-s + (2.88e13 − 2.15e13i)15-s + 7.03e13·16-s + 6.60e14i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.598 − 0.801i)3-s − 0.500·4-s + 0.277i·5-s + (0.566 − 0.423i)6-s − 1.74·7-s − 0.353i·8-s + (−0.284 + 0.958i)9-s − 0.196·10-s − 1.27i·11-s + (0.299 + 0.400i)12-s + 1.14·13-s − 1.23i·14-s + (0.222 − 0.165i)15-s + 0.250·16-s + 1.13i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{25}{2})\) |
\(\approx\) |
\(0.942450 + 0.313018i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.942450 + 0.313018i\) |
\(L(13)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.89e3iT \) |
| 3 | \( 1 + (3.17e5 + 4.25e5i)T \) |
good | 5 | \( 1 - 6.76e7iT - 5.96e16T^{2} \) |
| 7 | \( 1 + 2.41e10T + 1.91e20T^{2} \) |
| 11 | \( 1 + 4.00e12iT - 9.84e24T^{2} \) |
| 13 | \( 1 - 2.67e13T + 5.42e26T^{2} \) |
| 17 | \( 1 - 6.60e14iT - 3.39e29T^{2} \) |
| 19 | \( 1 + 3.05e14T + 4.89e30T^{2} \) |
| 23 | \( 1 - 7.66e14iT - 4.80e32T^{2} \) |
| 29 | \( 1 - 4.00e17iT - 1.25e35T^{2} \) |
| 31 | \( 1 - 1.08e18T + 6.20e35T^{2} \) |
| 37 | \( 1 - 2.67e18T + 4.33e37T^{2} \) |
| 41 | \( 1 + 3.10e19iT - 5.09e38T^{2} \) |
| 43 | \( 1 + 2.43e19T + 1.59e39T^{2} \) |
| 47 | \( 1 - 3.79e19iT - 1.35e40T^{2} \) |
| 53 | \( 1 - 8.82e19iT - 2.41e41T^{2} \) |
| 59 | \( 1 + 1.88e21iT - 3.16e42T^{2} \) |
| 61 | \( 1 + 1.92e21T + 7.04e42T^{2} \) |
| 67 | \( 1 + 7.96e21T + 6.69e43T^{2} \) |
| 71 | \( 1 - 2.04e22iT - 2.69e44T^{2} \) |
| 73 | \( 1 - 3.28e22T + 5.24e44T^{2} \) |
| 79 | \( 1 - 5.13e21T + 3.49e45T^{2} \) |
| 83 | \( 1 + 1.14e21iT - 1.14e46T^{2} \) |
| 89 | \( 1 - 3.69e23iT - 6.10e46T^{2} \) |
| 97 | \( 1 - 1.10e24T + 4.81e47T^{2} \) |
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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
−16.84379395263954427402307844880, −15.90156874703210938614560529320, −13.71767763370649105117867881516, −12.70830870422816203963260997921, −10.67251086298437860719355757207, −8.562799986798889794259426651545, −6.64438169167547012397760112823, −5.94896674554765800587462846112, −3.32089522526169056143070494622, −0.76899749529298679323563685936,
0.61567017855446826661169655473, 3.06021780283389970378503335732, 4.51683358311718038156666834832, 6.37252564123181990505634955493, 9.280003670636718388586550580495, 10.18636110977645162462806914311, 11.91575273344811772176171136136, 13.20838237916596027399257910963, 15.49241698686138553085991824011, 16.65174454861606567790148718490