L(s) = 1 | − 1.82i·2-s − 1.32·4-s + (−1.94 + 1.09i)5-s + 1.45i·7-s − 1.23i·8-s + (1.99 + 3.55i)10-s + 3.89·11-s − 3.05i·13-s + 2.64·14-s − 4.89·16-s − 3.92i·17-s + 19-s + (2.57 − 1.45i)20-s − 7.10i·22-s − 5.37i·23-s + ⋯ |
L(s) = 1 | − 1.28i·2-s − 0.660·4-s + (−0.871 + 0.490i)5-s + 0.548i·7-s − 0.437i·8-s + (0.632 + 1.12i)10-s + 1.17·11-s − 0.848i·13-s + 0.706·14-s − 1.22·16-s − 0.951i·17-s + 0.229·19-s + (0.575 − 0.324i)20-s − 1.51i·22-s − 1.12i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.314200 - 1.19806i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.314200 - 1.19806i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.94 - 1.09i)T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 1.82iT - 2T^{2} \) |
| 7 | \( 1 - 1.45iT - 7T^{2} \) |
| 11 | \( 1 - 3.89T + 11T^{2} \) |
| 13 | \( 1 + 3.05iT - 13T^{2} \) |
| 17 | \( 1 + 3.92iT - 17T^{2} \) |
| 23 | \( 1 + 5.37iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 + 5.95iT - 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.45iT - 43T^{2} \) |
| 47 | \( 1 + 4.90iT - 47T^{2} \) |
| 53 | \( 1 + 4.23iT - 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 9.84iT - 67T^{2} \) |
| 71 | \( 1 + 8.64T + 71T^{2} \) |
| 73 | \( 1 - 2.43iT - 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 12.6iT - 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.05iT - 97T^{2} \) |
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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
−10.07307769870945615413088265411, −9.151772883828519452436077111494, −8.396343213704546878265162724776, −7.18609442448184611324665025435, −6.52469126659464888191114483553, −5.10482415866519286995933906796, −3.94943107913009062109312888964, −3.22589175694575022491115138366, −2.25204804917240107294739616917, −0.64216144947282649167307904214,
1.50544756630966692402952392224, 3.61866432742605536984680181024, 4.37151454589885872487660082684, 5.36387945295630377213066763438, 6.47132997318405382993749358085, 7.06697180398603449679906295198, 7.83364885924227132322840395432, 8.673695333196501433742134946850, 9.244839906639423850939284890628, 10.50040913609446747705890575016