L(s) = 1 | + (−1.70 + 0.707i)3-s + (0.292 + 0.707i)5-s + (−1.70 + 4.12i)7-s + (0.292 − 0.292i)9-s + (1.70 + 0.707i)11-s + (−1 − 0.999i)15-s + (1 + 4i)17-s + (−4.41 − 4.41i)19-s − 8.24i·21-s + (4.53 + 1.87i)23-s + (3.12 − 3.12i)25-s + (1.82 − 4.41i)27-s + (−0.878 − 2.12i)29-s + (5.12 − 2.12i)31-s − 3.41·33-s + ⋯ |
L(s) = 1 | + (−0.985 + 0.408i)3-s + (0.130 + 0.316i)5-s + (−0.645 + 1.55i)7-s + (0.0976 − 0.0976i)9-s + (0.514 + 0.213i)11-s + (−0.258 − 0.258i)15-s + (0.242 + 0.970i)17-s + (−1.01 − 1.01i)19-s − 1.79i·21-s + (0.945 + 0.391i)23-s + (0.624 − 0.624i)25-s + (0.351 − 0.849i)27-s + (−0.163 − 0.393i)29-s + (0.919 − 0.381i)31-s − 0.594·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.445351 + 0.543824i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.445351 + 0.543824i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-1 - 4i)T \) |
good | 3 | \( 1 + (1.70 - 0.707i)T + (2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.292 - 0.707i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (1.70 - 4.12i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.70 - 0.707i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 19 | \( 1 + (4.41 + 4.41i)T + 19iT^{2} \) |
| 23 | \( 1 + (-4.53 - 1.87i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (0.878 + 2.12i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (-5.12 + 2.12i)T + (21.9 - 21.9i)T^{2} \) |
| 37 | \( 1 + (-1.70 + 0.707i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (3.70 - 8.94i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (1.24 - 1.24i)T - 43iT^{2} \) |
| 47 | \( 1 - 7.17iT - 47T^{2} \) |
| 53 | \( 1 + (-7.82 - 7.82i)T + 53iT^{2} \) |
| 59 | \( 1 + (-8.41 + 8.41i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.87 - 11.7i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 + (-2.29 + 0.949i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (5.36 + 12.9i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-12.5 - 5.19i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.24 - 1.24i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.65iT - 89T^{2} \) |
| 97 | \( 1 + (-0.778 - 1.87i)T + (-68.5 + 68.5i)T^{2} \) |
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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
−13.25100858614872392796178439056, −12.30654620011648464313181417029, −11.50212147676183983148593346576, −10.55329396069843070742757319824, −9.449042012786102215074637020806, −8.451742630397633008855426735450, −6.53538641281901950419130297654, −5.91186569801768685588346898984, −4.64608879087912381497830631296, −2.69085940400383342724202037154,
0.844277900644065299776300647648, 3.65744991360970000038252171712, 5.14431452819139227153559737165, 6.53172030298454632147526814749, 7.14122774594751221110520638400, 8.762924745979933267427006587083, 10.09383025157314976992291234179, 10.90042371237740653021200544863, 11.97065692342899570696881108481, 12.87070555189741735501339467034