L(s) = 1 | + (0.694 − 3.93i)2-s + (−2.84 − 16.1i)3-s + (−15.0 − 5.47i)4-s + (−37.4 + 31.4i)5-s − 65.4·6-s + (−21.7 + 18.2i)7-s + (−32 + 55.4i)8-s + (−23.5 + 8.58i)9-s + (97.7 + 169. i)10-s + (−281. + 487. i)11-s + (−45.4 + 257. i)12-s + (−128. − 46.7i)13-s + (56.8 + 98.5i)14-s + (613. + 514. i)15-s + (196. + 164. i)16-s + (673. − 245. i)17-s + ⋯ |
L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.182 − 1.03i)3-s + (−0.469 − 0.171i)4-s + (−0.669 + 0.562i)5-s − 0.742·6-s + (−0.168 + 0.141i)7-s + (−0.176 + 0.306i)8-s + (−0.0970 + 0.0353i)9-s + (0.309 + 0.535i)10-s + (−0.702 + 1.21i)11-s + (−0.0911 + 0.517i)12-s + (−0.210 − 0.0766i)13-s + (0.0775 + 0.134i)14-s + (0.703 + 0.590i)15-s + (0.191 + 0.160i)16-s + (0.565 − 0.205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.373831 + 0.232847i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.373831 + 0.232847i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.694 + 3.93i)T \) |
| 37 | \( 1 + (6.74e3 - 4.88e3i)T \) |
good | 3 | \( 1 + (2.84 + 16.1i)T + (-228. + 83.1i)T^{2} \) |
| 5 | \( 1 + (37.4 - 31.4i)T + (542. - 3.07e3i)T^{2} \) |
| 7 | \( 1 + (21.7 - 18.2i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (281. - 487. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (128. + 46.7i)T + (2.84e5 + 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-673. + 245. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 19 | \( 1 + (166. + 943. i)T + (-2.32e6 + 8.46e5i)T^{2} \) |
| 23 | \( 1 + (-1.45e3 - 2.51e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (4.24e3 - 7.35e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + 2.02e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (3.26e3 + 1.18e3i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 + 3.86e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.98e3 + 1.21e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (6.09e3 + 5.11e3i)T + (7.26e7 + 4.11e8i)T^{2} \) |
| 59 | \( 1 + (5.75e3 + 4.82e3i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (4.10e4 + 1.49e4i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (984. - 825. i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-2.17e3 - 1.23e4i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 + 7.40e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-8.18e3 + 6.87e3i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-6.05e4 + 2.20e4i)T + (3.01e9 - 2.53e9i)T^{2} \) |
| 89 | \( 1 + (1.40e4 + 1.17e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-4.76e4 - 8.25e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.35604693756098200341611810122, −12.60717448035665100529518635425, −11.79114433762312929656732527091, −10.68130945577193903387659353029, −9.424489287741813590989883876248, −7.66371957055384884518791618935, −6.96526217101211039085017505866, −5.10604745626975919744201729256, −3.27526252618747049527229244791, −1.69198638401707796276777738792,
0.18997790836847177773618499255, 3.59301002808005022155404671505, 4.71107106113341713327201175616, 5.89144854153775110845451141073, 7.68470512119470363684900545068, 8.672182005008169521757063869873, 9.956302308165952547977041719902, 11.02551562171214743286636230714, 12.40506243334842110329583118245, 13.51678161103803596797136420202