| L(s) = 1 | + (−17.2 + 3.03i)3-s + (−15.3 + 12.9i)5-s + (−26.4 + 45.7i)7-s + (211. − 76.8i)9-s + (−106. − 184. i)11-s + (67.2 + 11.8i)13-s + (225. − 269. i)15-s + (86.5 + 31.4i)17-s + (360. + 10.5i)19-s + (316. − 868. i)21-s + (152. + 128. i)23-s + (−38.3 + 217. i)25-s + (−2.17e3 + 1.25e3i)27-s + (−125. − 345. i)29-s + (−715. − 412. i)31-s + ⋯ |
| L(s) = 1 | + (−1.91 + 0.337i)3-s + (−0.615 + 0.516i)5-s + (−0.539 + 0.934i)7-s + (2.60 − 0.948i)9-s + (−0.880 − 1.52i)11-s + (0.398 + 0.0702i)13-s + (1.00 − 1.19i)15-s + (0.299 + 0.108i)17-s + (0.999 + 0.0293i)19-s + (0.716 − 1.96i)21-s + (0.288 + 0.242i)23-s + (−0.0614 + 0.348i)25-s + (−2.98 + 1.72i)27-s + (−0.149 − 0.411i)29-s + (−0.744 − 0.429i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.396345 - 0.178878i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.396345 - 0.178878i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-360. - 10.5i)T \) |
| good | 3 | \( 1 + (17.2 - 3.03i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (15.3 - 12.9i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (26.4 - 45.7i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (106. + 184. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-67.2 - 11.8i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-86.5 - 31.4i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-152. - 128. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (125. + 345. i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (715. + 412. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 2.46e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.69e3 + 299. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (1.34e3 - 1.12e3i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-1.84e3 + 670. i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-1.28e3 + 1.52e3i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-836. + 2.29e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (1.06e3 + 891. i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-415. - 1.14e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (5.93e3 + 7.07e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (258. + 1.46e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-8.75e3 + 1.54e3i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-3.71e3 + 6.44e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (4.04e3 + 712. i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (2.46e3 - 6.77e3i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−13.27687621704276895577367326763, −12.20183486432766955890359132799, −11.29106057572985082936237042115, −10.75544852239902807110224407946, −9.355225053432475660462657186869, −7.48562783563535090106268277365, −6.02097845567805496531838450534, −5.44567366938827743391606146693, −3.53755564761111289891208541145, −0.37686698921318365416916284022,
1.00265052177866970043513434296, 4.33353993277139490188600768409, 5.33211403022657548867957869665, 6.86873302148017017891169677116, 7.59723636663417624761417396937, 9.923349895331728224087853796110, 10.65820533241839258589377358921, 11.87513818638315310376421928607, 12.57658853452028452241508389347, 13.39787860880583836538737933393
