L(s) = 1 | + (0.707 − 0.707i)2-s + (0.717 + 1.57i)3-s − 1.00i·4-s + (0.882 + 3.29i)5-s + (1.62 + 0.607i)6-s + (−2.36 + 1.18i)7-s + (−0.707 − 0.707i)8-s + (−1.96 + 2.26i)9-s + (2.95 + 1.70i)10-s + (−2.62 + 0.704i)11-s + (1.57 − 0.717i)12-s + (0.963 − 3.47i)13-s + (−0.835 + 2.51i)14-s + (−4.55 + 3.75i)15-s − 1.00·16-s + 5.38·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.414 + 0.910i)3-s − 0.500i·4-s + (0.394 + 1.47i)5-s + (0.662 + 0.247i)6-s + (−0.894 + 0.447i)7-s + (−0.250 − 0.250i)8-s + (−0.656 + 0.754i)9-s + (0.933 + 0.539i)10-s + (−0.792 + 0.212i)11-s + (0.455 − 0.207i)12-s + (0.267 − 0.963i)13-s + (−0.223 + 0.670i)14-s + (−1.17 + 0.969i)15-s − 0.250·16-s + 1.30·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36321 + 1.28015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36321 + 1.28015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.717 - 1.57i)T \) |
| 7 | \( 1 + (2.36 - 1.18i)T \) |
| 13 | \( 1 + (-0.963 + 3.47i)T \) |
good | 5 | \( 1 + (-0.882 - 3.29i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (2.62 - 0.704i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 - 5.38T + 17T^{2} \) |
| 19 | \( 1 + (1.50 - 5.63i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 + (-7.19 + 4.15i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.302 + 1.12i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (3.35 - 3.35i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.36 - 0.366i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-8.98 - 5.18i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-12.4 + 3.34i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.31 + 1.91i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.350 + 0.350i)T + 59iT^{2} \) |
| 61 | \( 1 + (-6.12 - 10.6i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.546 - 0.146i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.45 - 9.16i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (13.2 + 3.53i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.696 - 1.20i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.65 + 2.65i)T - 83iT^{2} \) |
| 89 | \( 1 + (9.70 + 9.70i)T + 89iT^{2} \) |
| 97 | \( 1 + (-12.8 + 3.44i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.61033685411255249397400051275, −10.16286663248179379773269393043, −9.892267673481739676585455605336, −8.452684904681742171860799576323, −7.43056597866230070152551947250, −5.98024629705701827967648739241, −5.63803919901951024914773249521, −3.99902315157725792019874586489, −3.03640176388744264589777284925, −2.55835897549011906600336543369,
0.890690037406920705913730865527, 2.57399018109276624745467231732, 3.90740287194213669582836473135, 5.12003016360530043662271639350, 6.01060884211834219719933994952, 6.94310545283340147199256521831, 7.80917427631279201755656186600, 8.835658430056179461744967454146, 9.234180746940816161923545253315, 10.55705233913226594140567598532