L(s) = 1 | − 1.19e3·5-s + 1.84e4·7-s − 1.35e5·11-s − 8.48e5·13-s + 7.12e6·17-s − 5.04e6·19-s + 1.48e7·23-s − 4.74e7·25-s + 1.15e8·29-s − 1.63e8·31-s − 2.19e7·35-s − 2.23e8·37-s − 1.05e8·41-s + 1.41e9·43-s − 2.46e9·47-s − 1.63e9·49-s + 4.83e8·53-s + 1.61e8·55-s − 6.15e9·59-s − 7.53e9·61-s + 1.00e9·65-s − 8.76e9·67-s + 1.04e10·71-s − 3.17e10·73-s − 2.51e9·77-s − 3.98e10·79-s − 1.35e10·83-s + ⋯ |
L(s) = 1 | − 0.170·5-s + 0.415·7-s − 0.254·11-s − 0.633·13-s + 1.21·17-s − 0.467·19-s + 0.482·23-s − 0.970·25-s + 1.04·29-s − 1.02·31-s − 0.0707·35-s − 0.530·37-s − 0.142·41-s + 1.47·43-s − 1.57·47-s − 0.827·49-s + 0.158·53-s + 0.0433·55-s − 1.12·59-s − 1.14·61-s + 0.107·65-s − 0.793·67-s + 0.684·71-s − 1.79·73-s − 0.105·77-s − 1.45·79-s − 0.376·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 238 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 2640 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 135884 T + p^{11} T^{2} \) |
| 13 | \( 1 + 848186 T + p^{11} T^{2} \) |
| 17 | \( 1 - 7124606 T + p^{11} T^{2} \) |
| 19 | \( 1 + 5046316 T + p^{11} T^{2} \) |
| 23 | \( 1 - 14891224 T + p^{11} T^{2} \) |
| 29 | \( 1 - 115001346 T + p^{11} T^{2} \) |
| 31 | \( 1 + 163990552 T + p^{11} T^{2} \) |
| 37 | \( 1 + 223622178 T + p^{11} T^{2} \) |
| 41 | \( 1 + 105358314 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1419475852 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2469276960 T + p^{11} T^{2} \) |
| 53 | \( 1 - 483704986 T + p^{11} T^{2} \) |
| 59 | \( 1 + 6151842476 T + p^{11} T^{2} \) |
| 61 | \( 1 + 7532732282 T + p^{11} T^{2} \) |
| 67 | \( 1 + 8764949068 T + p^{11} T^{2} \) |
| 71 | \( 1 - 10401627752 T + p^{11} T^{2} \) |
| 73 | \( 1 + 31738391270 T + p^{11} T^{2} \) |
| 79 | \( 1 + 39880016072 T + p^{11} T^{2} \) |
| 83 | \( 1 + 13513323988 T + p^{11} T^{2} \) |
| 89 | \( 1 + 81514517226 T + p^{11} T^{2} \) |
| 97 | \( 1 - 30783027074 T + p^{11} T^{2} \) |
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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
−11.84812196191024582740870245321, −10.68192945597756813832942610858, −9.592505180442608584531936503029, −8.235491195221083297853121371203, −7.27043119287031754576630458666, −5.74419682716315104183926686585, −4.53572009400874848739329284462, −3.05064459256594690459725398210, −1.55300842706871748756266367622, 0,
1.55300842706871748756266367622, 3.05064459256594690459725398210, 4.53572009400874848739329284462, 5.74419682716315104183926686585, 7.27043119287031754576630458666, 8.235491195221083297853121371203, 9.592505180442608584531936503029, 10.68192945597756813832942610858, 11.84812196191024582740870245321