| L(s) = 1 | + (0.0336 − 0.234i)2-s + (−0.415 − 0.909i)3-s + (1.86 + 0.547i)4-s + (−0.788 − 0.909i)5-s + (−0.226 + 0.0666i)6-s + (0.0566 + 0.0363i)7-s + (0.387 − 0.848i)8-s + (−0.654 + 0.755i)9-s + (−0.239 + 0.153i)10-s + (0.272 + 1.89i)11-s + (−0.276 − 1.92i)12-s + (−3.64 + 2.34i)13-s + (0.0104 − 0.0120i)14-s + (−0.499 + 1.09i)15-s + (3.08 + 1.98i)16-s + (−6.53 + 1.92i)17-s + ⋯ |
| L(s) = 1 | + (0.0237 − 0.165i)2-s + (−0.239 − 0.525i)3-s + (0.932 + 0.273i)4-s + (−0.352 − 0.406i)5-s + (−0.0926 + 0.0271i)6-s + (0.0214 + 0.0137i)7-s + (0.136 − 0.299i)8-s + (−0.218 + 0.251i)9-s + (−0.0757 + 0.0486i)10-s + (0.0820 + 0.570i)11-s + (−0.0798 − 0.555i)12-s + (−1.01 + 0.649i)13-s + (0.00278 − 0.00321i)14-s + (−0.129 + 0.282i)15-s + (0.771 + 0.495i)16-s + (−1.58 + 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.914972 - 0.253265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.914972 - 0.253265i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-4.60 + 1.32i)T \) |
| good | 2 | \( 1 + (-0.0336 + 0.234i)T + (-1.91 - 0.563i)T^{2} \) |
| 5 | \( 1 + (0.788 + 0.909i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.0566 - 0.0363i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.272 - 1.89i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.64 - 2.34i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (6.53 - 1.92i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.374 + 0.110i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.99 + 0.584i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 3.40i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (2.17 - 2.50i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.92 - 3.37i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.07 + 8.92i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + (6.34 + 4.07i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (5.88 - 3.78i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.54 - 7.77i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.355 - 2.47i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.74 - 12.1i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (4.64 + 1.36i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-8.33 + 5.35i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-8.44 + 9.74i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (5.92 + 12.9i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-11.9 - 13.7i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−14.82173496527835891152435834469, −13.25711972541375393389499101793, −12.29142640299957867876253622502, −11.60121250932827244175275542061, −10.41360296393515482482134075548, −8.752196415028187000656501119093, −7.37307355716679392670133321698, −6.49416561122914893266166048492, −4.52500105086563036338449452214, −2.26496102753342117966133774533,
2.94217272145115175623566715862, 4.99675200914753326960356661409, 6.46653449850347498304892020960, 7.57883878209062866228617005013, 9.241235548347207714824572394339, 10.71696801337714522051630570485, 11.19809827088956362098649313337, 12.44964552584518963819163505056, 14.04822771914891263263570756251, 15.21177236221531310089655728044
