L(s) = 1 | + (6.89e4 − 6.89e4i)2-s + (5.43e7 + 5.43e7i)3-s − 5.20e9i·4-s + (1.52e11 − 1.20e10i)5-s + 7.48e12·6-s + (3.40e13 − 3.40e13i)7-s + (−6.30e13 − 6.30e13i)8-s + 4.04e15i·9-s + (9.65e15 − 1.13e16i)10-s − 2.53e16·11-s + (2.82e17 − 2.82e17i)12-s + (−3.66e17 − 3.66e17i)13-s − 4.69e18i·14-s + (8.91e18 + 7.60e18i)15-s + 1.36e19·16-s + (−5.02e19 + 5.02e19i)17-s + ⋯ |
L(s) = 1 | + (1.05 − 1.05i)2-s + (1.26 + 1.26i)3-s − 1.21i·4-s + (0.996 − 0.0787i)5-s + 2.65·6-s + (1.02 − 1.02i)7-s + (−0.224 − 0.224i)8-s + 2.18i·9-s + (0.965 − 1.13i)10-s − 0.551·11-s + (1.53 − 1.53i)12-s + (−0.551 − 0.551i)13-s − 2.15i·14-s + (1.35 + 1.15i)15-s + 0.741·16-s + (−1.03 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(6.783146398\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.783146398\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.52e11 + 1.20e10i)T \) |
good | 2 | \( 1 + (-6.89e4 + 6.89e4i)T - 4.29e9iT^{2} \) |
| 3 | \( 1 + (-5.43e7 - 5.43e7i)T + 1.85e15iT^{2} \) |
| 7 | \( 1 + (-3.40e13 + 3.40e13i)T - 1.10e27iT^{2} \) |
| 11 | \( 1 + 2.53e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (3.66e17 + 3.66e17i)T + 4.42e35iT^{2} \) |
| 17 | \( 1 + (5.02e19 - 5.02e19i)T - 2.36e39iT^{2} \) |
| 19 | \( 1 + 6.86e19iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (1.56e21 + 1.56e21i)T + 3.76e43iT^{2} \) |
| 29 | \( 1 + 6.33e22iT - 6.26e46T^{2} \) |
| 31 | \( 1 - 5.37e23T + 5.29e47T^{2} \) |
| 37 | \( 1 + (4.80e24 - 4.80e24i)T - 1.52e50iT^{2} \) |
| 41 | \( 1 + 7.84e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + (-6.30e25 - 6.30e25i)T + 1.86e52iT^{2} \) |
| 47 | \( 1 + (-2.58e26 + 2.58e26i)T - 3.21e53iT^{2} \) |
| 53 | \( 1 + (4.72e27 + 4.72e27i)T + 1.50e55iT^{2} \) |
| 59 | \( 1 - 2.35e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 + 2.89e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (6.00e28 - 6.00e28i)T - 2.71e58iT^{2} \) |
| 71 | \( 1 + 3.68e28T + 1.73e59T^{2} \) |
| 73 | \( 1 + (-1.90e29 - 1.90e29i)T + 4.22e59iT^{2} \) |
| 79 | \( 1 - 3.04e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (-1.94e30 - 1.94e30i)T + 2.57e61iT^{2} \) |
| 89 | \( 1 + 2.88e31iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (1.59e31 - 1.59e31i)T - 3.77e63iT^{2} \) |
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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
−15.07409785052187230400209600743, −14.06373226880783867186080028916, −13.24762349289070477553042193604, −10.77283818861638466600968669958, −10.09543540935914333069176152768, −8.246503438705745099738645633983, −5.01952984003516179309867492833, −4.19943226620847429675390078995, −2.79134754662911298419299185633, −1.75946380887577114920208414982,
1.72873055876823573074643766867, 2.69347141804236883324572964102, 5.00482090463608627664072513311, 6.44413544748130003591270970868, 7.66888482799622440613002222427, 9.013674431654323688352927917511, 12.29921052768529585686189974372, 13.60694583676260591143980746200, 14.23923892939614306604343857163, 15.34285233205308040611395535373