L(s) = 1 | − 2.04e3·4-s − 7.68e4·7-s − 2.24e6·13-s + 4.19e6·16-s + 2.05e7·19-s + 1.57e8·28-s + 2.96e8·31-s − 7.82e8·37-s + 1.76e9·43-s + 3.93e9·49-s + 4.60e9·52-s − 1.29e10·61-s − 8.58e9·64-s + 5.95e9·67-s + 1.98e10·73-s − 4.21e10·76-s + 3.28e10·79-s + 1.72e11·91-s − 5.29e10·97-s − 7.30e10·103-s + 2.57e9·109-s − 3.22e11·112-s + ⋯ |
L(s) = 1 | − 4-s − 1.72·7-s − 1.67·13-s + 16-s + 1.90·19-s + 1.72·28-s + 1.85·31-s − 1.85·37-s + 1.83·43-s + 1.98·49-s + 1.67·52-s − 1.96·61-s − 64-s + 0.538·67-s + 1.11·73-s − 1.90·76-s + 1.20·79-s + 2.90·91-s − 0.626·97-s − 0.621·103-s + 0.0160·109-s − 1.72·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + p^{11} T^{2} \) |
| 7 | \( 1 + 76885 T + p^{11} T^{2} \) |
| 11 | \( 1 + p^{11} T^{2} \) |
| 13 | \( 1 + 2248615 T + p^{11} T^{2} \) |
| 17 | \( 1 + p^{11} T^{2} \) |
| 19 | \( 1 - 20581901 T + p^{11} T^{2} \) |
| 23 | \( 1 + p^{11} T^{2} \) |
| 29 | \( 1 + p^{11} T^{2} \) |
| 31 | \( 1 - 296476943 T + p^{11} T^{2} \) |
| 37 | \( 1 + 782919730 T + p^{11} T^{2} \) |
| 41 | \( 1 + p^{11} T^{2} \) |
| 43 | \( 1 - 1768358135 T + p^{11} T^{2} \) |
| 47 | \( 1 + p^{11} T^{2} \) |
| 53 | \( 1 + p^{11} T^{2} \) |
| 59 | \( 1 + p^{11} T^{2} \) |
| 61 | \( 1 + 12977292913 T + p^{11} T^{2} \) |
| 67 | \( 1 - 5951291615 T + p^{11} T^{2} \) |
| 71 | \( 1 + p^{11} T^{2} \) |
| 73 | \( 1 - 19805520230 T + p^{11} T^{2} \) |
| 79 | \( 1 - 32885832404 T + p^{11} T^{2} \) |
| 83 | \( 1 + p^{11} T^{2} \) |
| 89 | \( 1 + p^{11} T^{2} \) |
| 97 | \( 1 + 52968566635 T + p^{11} T^{2} \) |
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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
−9.653742775597079311787947864048, −9.201831781103005144958437215192, −7.78921119887625613461070244146, −6.86497096692146027521018895758, −5.65676522489681026646835491983, −4.72880786556123502603082850055, −3.49007796186114499842248587738, −2.71244369456791502437101869406, −0.851722723247240775556568612314, 0,
0.851722723247240775556568612314, 2.71244369456791502437101869406, 3.49007796186114499842248587738, 4.72880786556123502603082850055, 5.65676522489681026646835491983, 6.86497096692146027521018895758, 7.78921119887625613461070244146, 9.201831781103005144958437215192, 9.653742775597079311787947864048