| 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
−15.21177236221531310089655728044, −14.04822771914891263263570756251, −12.44964552584518963819163505056, −11.19809827088956362098649313337, −10.71696801337714522051630570485, −9.241235548347207714824572394339, −7.57883878209062866228617005013, −6.46653449850347498304892020960, −4.99675200914753326960356661409, −2.94217272145115175623566715862,
2.26496102753342117966133774533, 4.52500105086563036338449452214, 6.49416561122914893266166048492, 7.37307355716679392670133321698, 8.752196415028187000656501119093, 10.41360296393515482482134075548, 11.60121250932827244175275542061, 12.29142640299957867876253622502, 13.25711972541375393389499101793, 14.82173496527835891152435834469
