L(s) = 1 | + (0.965 + 0.258i)2-s + i·3-s + (0.866 + 0.499i)4-s + (2.30 − 0.618i)5-s + (−0.258 + 0.965i)6-s + (1.78 − 1.94i)7-s + (0.707 + 0.707i)8-s − 9-s + 2.38·10-s + (−1.45 − 1.45i)11-s + (−0.499 + 0.866i)12-s + (3.60 − 0.0570i)13-s + (2.23 − 1.42i)14-s + (0.618 + 2.30i)15-s + (0.500 + 0.866i)16-s + (−1.37 + 2.38i)17-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + 0.577i·3-s + (0.433 + 0.249i)4-s + (1.03 − 0.276i)5-s + (−0.105 + 0.394i)6-s + (0.676 − 0.736i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s + 0.755·10-s + (−0.439 − 0.439i)11-s + (−0.144 + 0.249i)12-s + (0.999 − 0.0158i)13-s + (0.596 − 0.379i)14-s + (0.159 + 0.595i)15-s + (0.125 + 0.216i)16-s + (−0.334 + 0.579i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.56430 + 0.542533i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.56430 + 0.542533i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (-1.78 + 1.94i)T \) |
| 13 | \( 1 + (-3.60 + 0.0570i)T \) |
good | 5 | \( 1 + (-2.30 + 0.618i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (1.45 + 1.45i)T + 11iT^{2} \) |
| 17 | \( 1 + (1.37 - 2.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.53 + 4.53i)T + 19iT^{2} \) |
| 23 | \( 1 + (-2.12 + 1.22i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.69 - 2.93i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.60 - 5.97i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (2.80 - 10.4i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (9.91 - 2.65i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.84 + 3.95i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0569 + 0.212i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.80 - 6.58i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.48 - 9.29i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + 8.92iT - 61T^{2} \) |
| 67 | \( 1 + (-4.57 + 4.57i)T - 67iT^{2} \) |
| 71 | \( 1 + (5.69 + 1.52i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (11.7 + 3.15i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.79 + 6.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.18 + 9.18i)T + 83iT^{2} \) |
| 89 | \( 1 + (-0.410 - 0.110i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.25 + 4.66i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.69848559543727674849786277514, −10.37022493547126790683039100297, −8.906282933596179404156905471570, −8.408398551215214738000079303964, −7.00063673828063017819707568241, −6.09339800988768912135499961381, −5.15868787793410818441478841317, −4.38565906445386191868198956675, −3.17990408696155100362064651742, −1.65857508314816451596022188427,
1.80205889466226978028946076199, 2.45154075540142861583721976465, 4.03517458942810182860619530279, 5.45402855336360709800931336568, 5.89174257599817684215504099735, 6.90268113740731142291173562804, 8.014048803808460695220683079489, 8.967582896769682019924973891682, 10.01226659174498117076866720073, 10.94466157418907328093570271238