| L(s) = 1 | − 2.23·2-s + 3.00·4-s + (1 + 2i)5-s + 2i·7-s − 2.23·8-s + (−2.23 − 4.47i)10-s − 4.47i·11-s − 4i·13-s − 4.47i·14-s − 0.999·16-s − 4.47·17-s + 4.47i·19-s + (3.00 + 6.00i)20-s + 10.0i·22-s + 6i·23-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s + (0.447 + 0.894i)5-s + 0.755i·7-s − 0.790·8-s + (−0.707 − 1.41i)10-s − 1.34i·11-s − 1.10i·13-s − 1.19i·14-s − 0.249·16-s − 1.08·17-s + 1.02i·19-s + (0.670 + 1.34i)20-s + 2.13i·22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2352769856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2352769856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 29 | \( 1 + (3 - 4.47i)T \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 31 | \( 1 + 4.47iT - 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 8.94iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 13.4iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−9.935738535811924421416301655404, −9.210825985635991103754745299279, −8.499357806274984232468047488467, −7.84748978019647332716369706895, −6.98626991785551425940130702943, −6.04387031888616029287739630217, −5.47331098420434753368379248904, −3.54740027338595141059816997852, −2.63801502454215859539058414259, −1.54946195645694292037376711820,
0.16288221280421833640861376902, 1.56097472786110788727540770295, 2.29797299929811916202021264823, 4.37214958392142835310739236006, 4.74544196587878280584759612203, 6.46890242925906042264335539151, 6.95726196215900363794146718354, 7.76450723859041965701026358941, 8.819287887331014677195795881103, 9.096349953110822791115792883863
