| L(s) = 1 | + (−5.21e3 − 9.03e3i)2-s + (−2.39e6 + 4.14e6i)3-s + (2.14e8 − 3.70e8i)4-s + (−6.54e9 − 1.13e10i)5-s + (4.98e10 + 3.81e−6i)6-s + (1.56e12 − 8.84e11i)7-s + (−1.00e13 − 0.000732i)8-s + (−1.14e13 − 1.98e13i)9-s + (−6.83e13 + 1.18e14i)10-s + (−3.73e14 + 6.47e14i)11-s + (1.02e15 + 1.77e15i)12-s + 1.37e16·13-s + (−1.61e16 − 9.48e15i)14-s + 6.26e16·15-s + (−6.24e16 − 1.08e17i)16-s + (1.73e17 − 3.00e17i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 − 0.389i)2-s + (−0.288 + 0.499i)3-s + (0.398 − 0.690i)4-s + (−0.479 − 0.831i)5-s + 0.259·6-s + (0.869 − 0.493i)7-s − 0.809·8-s + (−0.166 − 0.288i)9-s + (−0.216 + 0.374i)10-s + (−0.296 + 0.513i)11-s + (0.230 + 0.398i)12-s + 0.968·13-s + (−0.388 − 0.228i)14-s + 0.554·15-s + (−0.216 − 0.375i)16-s + (0.250 − 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0712i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.514090984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.514090984\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.39e6 - 4.14e6i)T \) |
| 7 | \( 1 + (-1.56e12 + 8.84e11i)T \) |
| good | 2 | \( 1 + (5.21e3 + 9.03e3i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (6.54e9 + 1.13e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (3.73e14 - 6.47e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 - 1.37e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-1.73e17 + 3.00e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.08e18 + 3.60e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (2.27e19 + 3.93e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 - 1.52e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-2.67e21 + 4.63e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (6.66e21 + 1.15e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 - 1.78e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 2.73e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-3.16e23 - 5.48e23i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (5.13e24 - 8.88e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-2.57e24 + 4.46e24i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (4.16e25 + 7.21e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (-1.90e26 + 3.30e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 - 1.14e27T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-8.24e24 + 1.42e25i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (5.91e26 + 1.02e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 - 2.16e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (5.27e27 + 9.13e27i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 8.93e28T + 4.13e57T^{2} \) |
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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
−11.27767907758653313361196188034, −10.53473605356265689122931038580, −9.231438672300708409347665301974, −8.046819787350474943425252562852, −6.41427406094960497169674287029, −5.01541853796475524286413311189, −4.23173670899426501404771747592, −2.43902257797653962914132656712, −1.06562736369914441791235937496, −0.40800492564540004971227048671,
1.36579020397556599026238816457, 2.68216608240611265283722693786, 3.79163052653220761725966713086, 5.70923311413396797055631468501, 6.69416251870285787761151684996, 7.916457124481477373722091589955, 8.477755949357483399162884675638, 10.71633121338275798656292335717, 11.54928963705634041568492948368, 12.51218612772114018065172695180
