| L(s) = 1 | − 516·3-s + 1.05e4·5-s − 4.93e4·7-s + 8.91e4·9-s − 3.09e5·11-s + 1.72e6·13-s − 5.43e6·15-s − 2.27e6·17-s + 4.55e6·19-s + 2.54e7·21-s + 7.28e6·23-s + 6.20e7·25-s + 4.54e7·27-s + 6.90e7·29-s + 1.41e8·31-s + 1.59e8·33-s − 5.19e8·35-s − 7.11e8·37-s − 8.89e8·39-s − 1.22e9·41-s − 3.36e7·43-s + 9.38e8·45-s − 1.23e8·47-s + 4.53e8·49-s + 1.17e9·51-s − 1.10e9·53-s − 3.25e9·55-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + 1.50·5-s − 1.10·7-s + 0.503·9-s − 0.579·11-s + 1.28·13-s − 1.84·15-s − 0.389·17-s + 0.421·19-s + 1.35·21-s + 0.235·23-s + 1.27·25-s + 0.609·27-s + 0.625·29-s + 0.889·31-s + 0.710·33-s − 1.67·35-s − 1.68·37-s − 1.57·39-s − 1.65·41-s − 0.0348·43-s + 0.758·45-s − 0.0783·47-s + 0.229·49-s + 0.477·51-s − 0.363·53-s − 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 172 p T + p^{11} T^{2} \) |
| 5 | \( 1 - 2106 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 49304 T + p^{11} T^{2} \) |
| 11 | \( 1 + 309420 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1723594 T + p^{11} T^{2} \) |
| 17 | \( 1 + 2279502 T + p^{11} T^{2} \) |
| 19 | \( 1 - 4550444 T + p^{11} T^{2} \) |
| 23 | \( 1 - 7282872 T + p^{11} T^{2} \) |
| 29 | \( 1 - 69040026 T + p^{11} T^{2} \) |
| 31 | \( 1 - 141740704 T + p^{11} T^{2} \) |
| 37 | \( 1 + 711366974 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1225262214 T + p^{11} T^{2} \) |
| 43 | \( 1 + 781540 p T + p^{11} T^{2} \) |
| 47 | \( 1 + 123214608 T + p^{11} T^{2} \) |
| 53 | \( 1 + 1106121582 T + p^{11} T^{2} \) |
| 59 | \( 1 + 9062779932 T + p^{11} T^{2} \) |
| 61 | \( 1 - 3854150458 T + p^{11} T^{2} \) |
| 67 | \( 1 + 15313764676 T + p^{11} T^{2} \) |
| 71 | \( 1 + 20619626328 T + p^{11} T^{2} \) |
| 73 | \( 1 + 2063718694 T + p^{11} T^{2} \) |
| 79 | \( 1 + 13689871472 T + p^{11} T^{2} \) |
| 83 | \( 1 - 65570428908 T + p^{11} T^{2} \) |
| 89 | \( 1 + 29715508854 T + p^{11} T^{2} \) |
| 97 | \( 1 + 23439626206 T + p^{11} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.06539113244063403649383866603, −10.70503098377332571558952312247, −10.04794069332455308134388525781, −8.809504873437101104602089273124, −6.63192379977397829682270075738, −6.06164478020120281337641536885, −5.08374053764501989704857028039, −3.04662885840744209085796412136, −1.40011838183417875178846971654, 0,
1.40011838183417875178846971654, 3.04662885840744209085796412136, 5.08374053764501989704857028039, 6.06164478020120281337641536885, 6.63192379977397829682270075738, 8.809504873437101104602089273124, 10.04794069332455308134388525781, 10.70503098377332571558952312247, 12.06539113244063403649383866603
