L(s) = 1 | − 2i·2-s + 2·3-s − 2·4-s + i·5-s − 4i·6-s + 9-s + 2·10-s − 2i·11-s − 4·12-s + (2 − 3i)13-s + 2i·15-s − 4·16-s + 6·17-s − 2i·18-s − 3i·19-s − 2i·20-s + ⋯ |
L(s) = 1 | − 1.41i·2-s + 1.15·3-s − 4-s + 0.447i·5-s − 1.63i·6-s + 0.333·9-s + 0.632·10-s − 0.603i·11-s − 1.15·12-s + (0.554 − 0.832i)13-s + 0.516i·15-s − 16-s + 1.45·17-s − 0.471i·18-s − 0.688i·19-s − 0.447i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01313 - 1.89306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01313 - 1.89306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2 + 3i)T \) |
good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - iT - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 3iT - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 3iT - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 11iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 14iT - 71T^{2} \) |
| 73 | \( 1 - 9iT - 73T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 - 11iT - 83T^{2} \) |
| 89 | \( 1 - 5iT - 89T^{2} \) |
| 97 | \( 1 + 9iT - 97T^{2} \) |
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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
−10.18803269324558568594717069969, −9.746101007605975765124687230574, −8.606407045003369079509021139899, −8.088788842302387893616119710109, −6.85266458704210460933678725820, −5.56138909409872972110325621622, −4.03358376185266144617444207717, −3.08957182611120525165091376776, −2.73064689323856064386024325161, −1.15690658711596644277879688543,
1.88813653210403727742605182787, 3.40767769512006132825402228318, 4.54523800628492560613720280741, 5.60119860321301535171771274622, 6.52707027617185373735316645633, 7.61738763057747751025397236972, 8.034339538881456639547981720508, 8.950676614316671573941948696336, 9.430037758233488900961057905407, 10.64234353303246160100060382416