| L(s) = 1 | + (4.55 − 2.62i)2-s + (−2.17 + 3.77i)4-s + (11.2 + 19.4i)5-s + (−61.1 + 114. i)7-s + 191. i·8-s + (102. + 59.1i)10-s + (204. + 118. i)11-s + 74.9i·13-s + (22.2 + 681. i)14-s + (432. + 749. i)16-s + (−706. + 1.22e3i)17-s + (1.90e3 − 1.09e3i)19-s − 98.0·20-s + 1.24e3·22-s + (−1.22e3 + 708. i)23-s + ⋯ |
| L(s) = 1 | + (0.804 − 0.464i)2-s + (−0.0680 + 0.117i)4-s + (0.201 + 0.348i)5-s + (−0.471 + 0.881i)7-s + 1.05i·8-s + (0.324 + 0.187i)10-s + (0.509 + 0.294i)11-s + 0.122i·13-s + (0.0303 + 0.928i)14-s + (0.422 + 0.732i)16-s + (−0.592 + 1.02i)17-s + (1.20 − 0.697i)19-s − 0.0548·20-s + 0.547·22-s + (−0.483 + 0.279i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.88031 + 1.17199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88031 + 1.17199i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (61.1 - 114. i)T \) |
| good | 2 | \( 1 + (-4.55 + 2.62i)T + (16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-11.2 - 19.4i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-204. - 118. i)T + (8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 74.9iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (706. - 1.22e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.90e3 + 1.09e3i)T + (1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.22e3 - 708. i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 2.67e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (18.6 + 10.7i)T + (1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.56e3 + 2.70e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.68e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.89e3 + 5.01e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-2.13e3 - 1.23e3i)T + (2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.11e4 + 1.93e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-4.47e4 + 2.58e4i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.28e4 + 3.95e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 7.73e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-6.30e3 - 3.64e3i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.77e4 - 8.27e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 9.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-6.62e4 - 1.14e5i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 5.85e4iT - 8.58e9T^{2} \) |
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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
−14.02988857286650791876011932423, −12.94596444090099604212795078115, −12.11582951000694331736460493921, −11.11868902713796621173158013011, −9.574155134794348531641979092916, −8.385865502635231747680095947146, −6.60127763289007128233539521514, −5.21303131392709208249752009191, −3.64982741814073211945824719114, −2.28547266182307013129577210315,
0.842456544254894539078996154675, 3.59144389550498865955422047260, 4.91855261287290290485846055918, 6.22885118838631929539861788206, 7.38649079681175034898518808529, 9.246326096366514884786693536921, 10.18500010770006919544461613763, 11.74792170048858959161748040640, 13.11786293889360692520733395576, 13.74281790519560995514914668562
