L(s) = 1 | + (−0.707 + 0.707i)3-s + (1.82 + 1.29i)5-s − 0.513i·7-s − 1.00i·9-s + (2.70 + 2.70i)11-s + (−3.45 + 1.03i)13-s + (−2.20 + 0.378i)15-s + (2.65 − 2.65i)17-s + (4.94 + 4.94i)19-s + (0.362 + 0.362i)21-s + (−1.20 − 1.20i)23-s + (1.66 + 4.71i)25-s + (0.707 + 0.707i)27-s − 3.33i·29-s + (−4.89 + 4.89i)31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.816 + 0.577i)5-s − 0.194i·7-s − 0.333i·9-s + (0.817 + 0.817i)11-s + (−0.957 + 0.287i)13-s + (−0.569 + 0.0977i)15-s + (0.643 − 0.643i)17-s + (1.13 + 1.13i)19-s + (0.0792 + 0.0792i)21-s + (−0.251 − 0.251i)23-s + (0.333 + 0.942i)25-s + (0.136 + 0.136i)27-s − 0.619i·29-s + (−0.879 + 0.879i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20775 + 0.926771i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20775 + 0.926771i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.82 - 1.29i)T \) |
| 13 | \( 1 + (3.45 - 1.03i)T \) |
good | 7 | \( 1 + 0.513iT - 7T^{2} \) |
| 11 | \( 1 + (-2.70 - 2.70i)T + 11iT^{2} \) |
| 17 | \( 1 + (-2.65 + 2.65i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.94 - 4.94i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.20 + 1.20i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.33iT - 29T^{2} \) |
| 31 | \( 1 + (4.89 - 4.89i)T - 31iT^{2} \) |
| 37 | \( 1 + 8.27iT - 37T^{2} \) |
| 41 | \( 1 + (8.18 - 8.18i)T - 41iT^{2} \) |
| 43 | \( 1 + (-7.27 - 7.27i)T + 43iT^{2} \) |
| 47 | \( 1 - 5.36iT - 47T^{2} \) |
| 53 | \( 1 + (0.721 - 0.721i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.69 - 8.69i)T - 59iT^{2} \) |
| 61 | \( 1 - 4.33T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + (-1.66 + 1.66i)T - 71iT^{2} \) |
| 73 | \( 1 + 5.73T + 73T^{2} \) |
| 79 | \( 1 - 5.21iT - 79T^{2} \) |
| 83 | \( 1 + 4.62iT - 83T^{2} \) |
| 89 | \( 1 + (-7.61 + 7.61i)T - 89iT^{2} \) |
| 97 | \( 1 - 1.65T + 97T^{2} \) |
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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
−10.24999383299484351637835030611, −9.724017424112603931607872328866, −9.221922867910564896385662564786, −7.63257053232696896926305595011, −7.01823348297160563077421545721, −6.01219816615549627271514849667, −5.20272649563794534169106539913, −4.14272757047071797266874663254, −2.94140629228395631106122990537, −1.56213343385010488931710063589,
0.877653547594334165315608561232, 2.17611567887567938575661958331, 3.56815239625961343138705016398, 5.09078508105802290607714357864, 5.56632783991200674127673249087, 6.55443285890910149686279318104, 7.44909953390322464406476776001, 8.533498050746010120358732596018, 9.282508837531313355772354588095, 10.04285276066004440811979095245