L(s) = 1 | + (−1.43 + 0.830i)3-s + (7.27 + 4.20i)5-s + (3.99 − 5.74i)7-s + (−3.12 + 5.40i)9-s + (−2.60 − 4.51i)11-s + 4.88i·13-s − 13.9·15-s + (6.68 − 3.86i)17-s + (30.6 + 17.6i)19-s + (−0.979 + 11.5i)21-s + (−13.3 + 23.0i)23-s + (22.7 + 39.4i)25-s − 25.3i·27-s + 45.1·29-s + (−35.0 + 20.2i)31-s + ⋯ |
L(s) = 1 | + (−0.479 + 0.276i)3-s + (1.45 + 0.840i)5-s + (0.571 − 0.820i)7-s + (−0.346 + 0.600i)9-s + (−0.237 − 0.410i)11-s + 0.375i·13-s − 0.930·15-s + (0.393 − 0.227i)17-s + (1.61 + 0.929i)19-s + (−0.0466 + 0.551i)21-s + (−0.578 + 1.00i)23-s + (0.911 + 1.57i)25-s − 0.937i·27-s + 1.55·29-s + (−1.12 + 0.652i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.57713 + 0.700596i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57713 + 0.700596i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-3.99 + 5.74i)T \) |
good | 3 | \( 1 + (1.43 - 0.830i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-7.27 - 4.20i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (2.60 + 4.51i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 4.88iT - 169T^{2} \) |
| 17 | \( 1 + (-6.68 + 3.86i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-30.6 - 17.6i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (13.3 - 23.0i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 45.1T + 841T^{2} \) |
| 31 | \( 1 + (35.0 - 20.2i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (3.97 - 6.89i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 26.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 0.403T + 1.84e3T^{2} \) |
| 47 | \( 1 + (28.2 + 16.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (40.6 + 70.3i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (7.62 - 4.40i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (25.3 + 14.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-43.7 - 75.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 27.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-75.3 + 43.5i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-60.8 + 105. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 46.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (52.7 + 30.4i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 66.4iT - 9.40e3T^{2} \) |
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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
−11.89089850418738812921005441319, −10.99839938436030912287900540252, −10.26572535490944821083382380059, −9.641898326774746695970942218429, −8.071639401009979875928143169221, −7.00446541494676103355941376872, −5.79262236455444665806266971352, −5.10235678282367856985610684149, −3.28787261066127326565908040346, −1.68727972118057331974029908246,
1.16372000153567695624490027420, 2.62677627598154578256915258673, 4.93027613600147710645792687425, 5.58992268769783016246903144312, 6.47596837747628401255884237294, 8.050112833024013741768827573563, 9.137634060889836166694287739853, 9.733608186953220035562854392055, 11.01216613667186381564552023040, 12.18824836129026439125815103928