L(s) = 1 | + (0.160 − 0.0926i)2-s + (−0.982 + 1.70i)4-s + (−1.29 + 1.82i)5-s + (−1.66 − 0.959i)7-s + 0.734i·8-s + (−0.0396 + 0.412i)10-s + (−1.95 − 3.39i)11-s + (2.82 + 2.24i)13-s − 0.355·14-s + (−1.89 − 3.28i)16-s + (−2.88 − 1.66i)17-s + (0.645 − 1.11i)19-s + (−1.82 − 3.99i)20-s + (−0.628 − 0.362i)22-s + (−5.24 + 3.02i)23-s + ⋯ |
L(s) = 1 | + (0.113 − 0.0654i)2-s + (−0.491 + 0.851i)4-s + (−0.580 + 0.814i)5-s + (−0.628 − 0.362i)7-s + 0.259i·8-s + (−0.0125 + 0.130i)10-s + (−0.590 − 1.02i)11-s + (0.782 + 0.622i)13-s − 0.0950·14-s + (−0.474 − 0.821i)16-s + (−0.699 − 0.403i)17-s + (0.148 − 0.256i)19-s + (−0.407 − 0.894i)20-s + (−0.133 − 0.0773i)22-s + (−1.09 + 0.630i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0125926 - 0.0448101i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0125926 - 0.0448101i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.29 - 1.82i)T \) |
| 13 | \( 1 + (-2.82 - 2.24i)T \) |
good | 2 | \( 1 + (-0.160 + 0.0926i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.66 + 0.959i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.95 + 3.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.88 + 1.66i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.645 + 1.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.24 - 3.02i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.97 + 8.61i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.08T + 31T^{2} \) |
| 37 | \( 1 + (8.81 - 5.09i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.79 + 4.84i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.62 - 4.40i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.87iT - 47T^{2} \) |
| 53 | \( 1 + 7.52iT - 53T^{2} \) |
| 59 | \( 1 + (2.77 - 4.80i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.38 - 7.58i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.09 - 2.36i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.63 - 9.75i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 1.67iT - 73T^{2} \) |
| 79 | \( 1 - 0.965T + 79T^{2} \) |
| 83 | \( 1 - 11.4iT - 83T^{2} \) |
| 89 | \( 1 + (-1.78 - 3.08i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.05 + 4.64i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.41804022361599375352908298058, −10.43445628518269738723887144501, −9.420998072222491152974319532430, −8.447881414787016188948554565997, −7.73857294469900127590118415682, −6.82750226024708801595669269240, −5.86597672140475957759191015958, −4.30339968331420520090126447709, −3.60055504681196530608507527593, −2.68196195199064586873354116840,
0.02470412705689474356316776013, 1.77056878815900694035087830192, 3.60749906762189161135024136529, 4.63239171637271844051455516263, 5.46631200487961641789451412628, 6.36709651795153910739642470758, 7.57751799517330772387263906309, 8.643162772339145644293934293081, 9.197403541542183210474963501775, 10.24038315150173811984882064164