L(s) = 1 | − 2i·2-s − 7.80i·3-s − 4·4-s + (−4.43 − 10.2i)5-s − 15.6·6-s − 21.1i·7-s + 8i·8-s − 33.9·9-s + (−20.5 + 8.86i)10-s + 4.34·11-s + 31.2i·12-s − 43.4i·13-s − 42.3·14-s + (−80.1 + 34.6i)15-s + 16·16-s + 0.244i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.50i·3-s − 0.5·4-s + (−0.396 − 0.918i)5-s − 1.06·6-s − 1.14i·7-s + 0.353i·8-s − 1.25·9-s + (−0.649 + 0.280i)10-s + 0.119·11-s + 0.751i·12-s − 0.926i·13-s − 0.807·14-s + (−1.37 + 0.595i)15-s + 0.250·16-s + 0.00349i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.755750 + 1.14974i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755750 + 1.14974i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 + (4.43 + 10.2i)T \) |
| 23 | \( 1 - 23iT \) |
good | 3 | \( 1 + 7.80iT - 27T^{2} \) |
| 7 | \( 1 + 21.1iT - 343T^{2} \) |
| 11 | \( 1 - 4.34T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 0.244iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 91.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 79.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.7iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 297. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 69.2iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 237. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 938.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 31.0iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 995.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 807. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 410.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 287. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3iT - 9.12e5T^{2} \) |
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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
−11.43767072117011732280827639949, −10.25575581420062804716461319341, −9.079880522245175083665891132346, −7.77990621517204298379963739466, −7.50845144918712294430362150025, −5.92110778171509331898473630134, −4.56355032215327709575502864504, −3.11487397558523962846994299465, −1.30028673500394124075377874972, −0.65745135156391597569051978304,
2.87941180580225039374219773194, 4.03889649283400677213026669376, 5.14299539094342394902562109524, 6.17577597112174242966019841705, 7.40997756947473068162579731031, 8.715287870000887863972968920020, 9.447331594759316696239435095186, 10.27997449325336302476239617274, 11.40827192378456991901840312759, 12.08576113870439603602682825154