| L(s) = 1 | + (5.53 − 9.58i)3-s + (654. − 378. i)5-s + (−680. − 6.31e3i)7-s + (9.78e3 + 1.69e4i)9-s + (−5.09e4 − 2.94e4i)11-s − 1.56e4i·13-s − 8.37e3i·15-s + (−2.64e5 − 1.52e5i)17-s + (−2.86e4 − 4.96e4i)19-s + (−6.43e4 − 2.84e4i)21-s + (5.22e5 − 3.01e5i)23-s + (−6.90e5 + 1.19e6i)25-s + 4.34e5·27-s + 2.18e6·29-s + (−2.56e6 + 4.45e6i)31-s + ⋯ |
| L(s) = 1 | + (0.0394 − 0.0683i)3-s + (0.468 − 0.270i)5-s + (−0.107 − 0.994i)7-s + (0.496 + 0.860i)9-s + (−1.04 − 0.605i)11-s − 0.151i·13-s − 0.0426i·15-s + (−0.768 − 0.443i)17-s + (−0.0504 − 0.0873i)19-s + (−0.0721 − 0.0319i)21-s + (0.389 − 0.224i)23-s + (−0.353 + 0.612i)25-s + 0.157·27-s + 0.572·29-s + (−0.499 + 0.865i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2626380788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2626380788\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (680. + 6.31e3i)T \) |
| good | 3 | \( 1 + (-5.53 + 9.58i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-654. + 378. i)T + (9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (5.09e4 + 2.94e4i)T + (1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.56e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (2.64e5 + 1.52e5i)T + (5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.86e4 + 4.96e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-5.22e5 + 3.01e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 2.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (2.56e6 - 4.45e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (4.31e5 + 7.47e5i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 9.62e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 2.43e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-6.97e6 - 1.20e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (5.35e7 - 9.26e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (2.51e7 - 4.35e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (8.06e7 - 4.65e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.98e8 + 1.14e8i)T + (1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.20e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-2.01e7 - 1.16e7i)T + (2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.09e8 - 1.20e8i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 2.59e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (8.63e8 - 4.98e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.37e9iT - 7.60e17T^{2} \) |
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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
−10.88819142375024875472005266338, −10.46712524678027623063687877582, −9.150272273880238279922997260159, −7.892446447050154338074817939769, −6.98177044968751756065700609037, −5.48011647069711556640327274265, −4.44665381948491440569210027814, −2.85309259341218491570497542216, −1.44528205750303341017193538935, −0.06195728471062734097458081775,
1.81039674733382191217136542117, 2.89364476247031642162285174340, 4.49070469868600347350124875079, 5.81810526865856181025833613456, 6.78257258826777797566846487879, 8.193739420305409628878888867593, 9.379722395925325081461639420366, 10.12508065331487543077379728244, 11.40417048977293538919948790370, 12.54687633115610115922343610206
