L(s) = 1 | + (0.259 − 1.39i)2-s + (−1.86 − 0.720i)4-s + (0.173 + 1.31i)5-s + (0.0444 + 0.165i)7-s + (−1.48 + 2.40i)8-s + (1.87 + 0.100i)10-s + (0.877 − 0.673i)11-s + (2.54 − 3.32i)13-s + (0.242 − 0.0187i)14-s + (2.96 + 2.68i)16-s − 1.67·17-s + (3.90 − 1.61i)19-s + (0.626 − 2.58i)20-s + (−0.708 − 1.39i)22-s + (−1.65 − 0.443i)23-s + ⋯ |
L(s) = 1 | + (0.183 − 0.983i)2-s + (−0.932 − 0.360i)4-s + (0.0776 + 0.590i)5-s + (0.0167 + 0.0626i)7-s + (−0.525 + 0.850i)8-s + (0.594 + 0.0317i)10-s + (0.264 − 0.203i)11-s + (0.706 − 0.921i)13-s + (0.0646 − 0.00502i)14-s + (0.740 + 0.672i)16-s − 0.406·17-s + (0.895 − 0.370i)19-s + (0.140 − 0.578i)20-s + (−0.151 − 0.297i)22-s + (−0.344 − 0.0924i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0367 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0367 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14968 - 1.10816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14968 - 1.10816i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.259 + 1.39i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.173 - 1.31i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (-0.0444 - 0.165i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.877 + 0.673i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.54 + 3.32i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 + (-3.90 + 1.61i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (1.65 + 0.443i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-6.46 - 0.851i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-3.09 + 1.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.58 + 8.64i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.790 + 2.94i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (5.37 + 7.00i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-0.598 - 0.345i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.43 - 10.7i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (3.24 - 0.427i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (1.28 - 9.76i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (5.81 - 7.57i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (0.107 + 0.107i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.93 + 3.93i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.777 + 1.34i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.82 + 0.635i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-10.7 + 10.7i)T - 89iT^{2} \) |
| 97 | \( 1 + (6.90 - 11.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.36663160852125292254735912652, −9.190832345815988236118684113952, −8.561831186924186025566099638318, −7.48629326425394026697174255965, −6.28856149648994433383519356742, −5.47470098693924054846914561446, −4.34787759771707918017269177421, −3.30468741429956944135865867315, −2.49284205818778475553729606520, −0.912582160127559515095939494002,
1.25655251750395832094214404980, 3.23307807620446676479100831192, 4.41493164809294752550790207138, 4.99402019126309845043979103586, 6.24035351702115587744933030693, 6.72296149502825243482691049048, 7.911606715018152479875211548533, 8.532833164088728137712986965408, 9.358397163145315651832222072553, 10.01055370128800591259836384832