L(s) = 1 | + 0.773i·3-s + (−1.98 + 1.02i)5-s + 1.12·7-s + 2.40·9-s + 2.52i·11-s + (1.37 − 3.33i)13-s + (−0.789 − 1.53i)15-s − 0.870i·17-s + 6.27i·19-s + 0.870i·21-s + 2.81i·23-s + (2.91 − 4.06i)25-s + 4.17i·27-s − 0.176·29-s − 0.388i·31-s + ⋯ |
L(s) = 1 | + 0.446i·3-s + (−0.889 + 0.456i)5-s + 0.425·7-s + 0.800·9-s + 0.760i·11-s + (0.381 − 0.924i)13-s + (−0.203 − 0.397i)15-s − 0.211i·17-s + 1.43i·19-s + 0.189i·21-s + 0.586i·23-s + (0.583 − 0.812i)25-s + 0.803i·27-s − 0.0328·29-s − 0.0698i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0822 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0822 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.376343187\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.376343187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.98 - 1.02i)T \) |
| 13 | \( 1 + (-1.37 + 3.33i)T \) |
good | 3 | \( 1 - 0.773iT - 3T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 - 2.52iT - 11T^{2} \) |
| 17 | \( 1 + 0.870iT - 17T^{2} \) |
| 19 | \( 1 - 6.27iT - 19T^{2} \) |
| 23 | \( 1 - 2.81iT - 23T^{2} \) |
| 29 | \( 1 + 0.176T + 29T^{2} \) |
| 31 | \( 1 + 0.388iT - 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 - 7.25iT - 41T^{2} \) |
| 43 | \( 1 - 4.52iT - 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 - 9.83iT - 53T^{2} \) |
| 59 | \( 1 - 12.1iT - 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 + 4.37iT - 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 - 8.31T + 79T^{2} \) |
| 83 | \( 1 + 9.15T + 83T^{2} \) |
| 89 | \( 1 + 6.15iT - 89T^{2} \) |
| 97 | \( 1 + 1.55T + 97T^{2} \) |
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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
−10.21993508654487510550058695246, −9.507043229432597562405000474564, −8.272368703754795578301709876593, −7.71283944544189769069227295166, −6.97666848652372395161698441518, −5.82058671731720189285247085981, −4.72396893718589182671895123448, −3.98792637964825967860589514211, −3.07786339284439538302207026765, −1.46736168650280227653785685243,
0.68504282224356816901857194584, 1.99045470697162462484057264141, 3.53873025416977412005105642745, 4.40361564411617381348760457315, 5.23202098881326474499164614068, 6.62803540144462535086142353934, 7.10075580731650222488833748982, 8.173360271086279325281338450285, 8.666824402543102241713051395585, 9.556108843360190989230872115077