| L(s) = 1 | + (1.09 + 0.894i)2-s + (0.718 + 0.502i)3-s + (0.399 + 1.95i)4-s + (2.40 + 0.210i)5-s + (0.336 + 1.19i)6-s + (−0.0989 − 0.0571i)7-s + (−1.31 + 2.50i)8-s + (−0.763 − 2.09i)9-s + (2.44 + 2.37i)10-s + (0.0164 − 0.0613i)11-s + (−0.698 + 1.60i)12-s + (−2.44 − 3.49i)13-s + (−0.0573 − 0.151i)14-s + (1.61 + 1.35i)15-s + (−3.68 + 1.56i)16-s + (−3.74 − 1.36i)17-s + ⋯ |
| L(s) = 1 | + (0.774 + 0.632i)2-s + (0.414 + 0.290i)3-s + (0.199 + 0.979i)4-s + (1.07 + 0.0939i)5-s + (0.137 + 0.487i)6-s + (−0.0374 − 0.0216i)7-s + (−0.464 + 0.885i)8-s + (−0.254 − 0.698i)9-s + (0.772 + 0.752i)10-s + (0.00495 − 0.0184i)11-s + (−0.201 + 0.464i)12-s + (−0.678 − 0.969i)13-s + (−0.0153 − 0.0403i)14-s + (0.418 + 0.350i)15-s + (−0.920 + 0.391i)16-s + (−0.907 − 0.330i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93581 + 1.36795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93581 + 1.36795i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.09 - 0.894i)T \) |
| 19 | \( 1 + (-2.92 - 3.23i)T \) |
| good | 3 | \( 1 + (-0.718 - 0.502i)T + (1.02 + 2.81i)T^{2} \) |
| 5 | \( 1 + (-2.40 - 0.210i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (0.0989 + 0.0571i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0164 + 0.0613i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (2.44 + 3.49i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (3.74 + 1.36i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.883 + 1.05i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.691 - 1.48i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (3.21 - 5.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.852 - 0.852i)T + 37iT^{2} \) |
| 41 | \( 1 + (-9.23 + 1.62i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.32 - 0.203i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (2.69 - 0.979i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (0.241 + 2.75i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (0.0340 + 0.0730i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (7.07 - 0.619i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-4.23 + 9.08i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-6.41 - 7.64i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-9.16 + 1.61i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.96 - 11.1i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.845 + 3.15i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (6.98 + 1.23i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-16.5 - 6.03i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.25352740114498567664661243953, −11.02734214717983564703444571038, −9.832061745295792115655057586543, −9.074779544310232409541949796826, −7.989855618465402372118846420858, −6.83459485494096646057880975292, −5.89371239424467644602226254293, −5.01207082109217830934957878800, −3.56923379304720355475677404407, −2.49301515626893814905863236480,
1.86618083323796413586717334085, 2.65481952797545253860906783957, 4.36693447527666724725288512302, 5.39339859332970052814030777690, 6.39887686469212036831763721768, 7.52233609124951693344527446039, 9.154641231325755216797527975395, 9.583082458613932662369710391155, 10.82658103439186875271893356399, 11.49906619948007446731856613407
