L(s) = 1 | − 2.52i·2-s + (−1.68 + 0.396i)3-s − 4.37·4-s − 5-s + (1 + 4.25i)6-s + (−2 − 1.73i)7-s + 5.98i·8-s + (2.68 − 1.33i)9-s + 2.52i·10-s − 0.792i·11-s + (7.37 − 1.73i)12-s − 5.84i·13-s + (−4.37 + 5.04i)14-s + (1.68 − 0.396i)15-s + 6.37·16-s + 1.37·17-s + ⋯ |
L(s) = 1 | − 1.78i·2-s + (−0.973 + 0.228i)3-s − 2.18·4-s − 0.447·5-s + (0.408 + 1.73i)6-s + (−0.755 − 0.654i)7-s + 2.11i·8-s + (0.895 − 0.445i)9-s + 0.798i·10-s − 0.238i·11-s + (2.12 − 0.500i)12-s − 1.61i·13-s + (−1.16 + 1.34i)14-s + (0.435 − 0.102i)15-s + 1.59·16-s + 0.332·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.112916 + 0.458469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.112916 + 0.458469i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.68 - 0.396i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 2 | \( 1 + 2.52iT - 2T^{2} \) |
| 11 | \( 1 + 0.792iT - 11T^{2} \) |
| 13 | \( 1 + 5.84iT - 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 1.87iT - 23T^{2} \) |
| 29 | \( 1 + 4.25iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 - 7.37T + 47T^{2} \) |
| 53 | \( 1 - 8.51iT - 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 + 13.5iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.37T + 79T^{2} \) |
| 83 | \( 1 + 5.48T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 1.08iT - 97T^{2} \) |
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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
−12.70631712610674145897036296752, −12.09989699445251765178065884802, −10.93458024962415248135667705405, −10.37212443462685416312377822520, −9.534861148159323211795242140822, −7.81474163165324799231768887311, −5.89018559577584905006613241720, −4.34856076692309413846383883668, −3.25364081147517126800276509792, −0.62929941910092166589925501370,
4.36291639411519999466566134470, 5.53347030414673233935182508219, 6.64152000925777107924857093507, 7.23403233782024251737915850216, 8.732626688137741717505069837863, 9.701799807669477897436972653816, 11.43473495685161413381131248208, 12.49833106299680201734535382150, 13.49812062410364439725879543618, 14.65610240751441889920576565338