| L(s) = 1 | − 1.52e5·5-s + 4.78e6·9-s − 4.63e7·13-s + 7.86e8·17-s + 1.72e10·25-s + 1.98e10·29-s + 1.28e11·37-s − 3.89e11·41-s − 7.31e11·45-s + 6.78e11·49-s − 1.71e12·53-s + 6.00e12·61-s + 7.08e12·65-s − 6.74e12·73-s + 2.28e13·81-s − 1.20e14·85-s − 2.97e12·89-s + 1.47e14·97-s − 3.74e13·101-s + 2.15e14·109-s + 3.77e14·113-s − 2.21e14·117-s + ⋯ |
| L(s) = 1 | − 1.95·5-s + 9-s − 0.738·13-s + 1.91·17-s + 2.82·25-s + 1.15·29-s + 1.35·37-s − 1.99·41-s − 1.95·45-s + 49-s − 1.45·53-s + 1.90·61-s + 1.44·65-s − 0.610·73-s + 81-s − 3.75·85-s − 0.0671·89-s + 1.81·97-s − 0.349·101-s + 1.18·109-s + 1.60·113-s − 0.738·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(1.308921081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308921081\) |
| \(L(8)\) |
|
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 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 5 | \( 1 + 152886 T + p^{14} T^{2} \) |
| 7 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 11 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 13 | \( 1 + 46322630 T + p^{14} T^{2} \) |
| 17 | \( 1 - 786851490 T + p^{14} T^{2} \) |
| 19 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 23 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 29 | \( 1 - 19896480282 T + p^{14} T^{2} \) |
| 31 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 37 | \( 1 - 128202918410 T + p^{14} T^{2} \) |
| 41 | \( 1 + 389417726862 T + p^{14} T^{2} \) |
| 43 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 47 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 53 | \( 1 + 1711657125270 T + p^{14} T^{2} \) |
| 59 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 61 | \( 1 - 6001852736858 T + p^{14} T^{2} \) |
| 67 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 71 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 73 | \( 1 + 6748633251470 T + p^{14} T^{2} \) |
| 79 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 83 | \( ( 1 - p^{7} T )( 1 + p^{7} T ) \) |
| 89 | \( 1 + 2971010122158 T + p^{14} T^{2} \) |
| 97 | \( 1 - 147017184612610 T + p^{14} T^{2} \) |
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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
−15.66786874144147117978132179447, −14.63507563229165500020002978840, −12.56249063453334051453101661395, −11.74303211748106497918450787313, −10.12031867483186501998928159364, −8.108936289661534573838202358758, −7.17327402680311819721263451580, −4.67432988969660809053179724331, −3.37846084977597128846508968282, −0.809306913244096063449419902023,
0.809306913244096063449419902023, 3.37846084977597128846508968282, 4.67432988969660809053179724331, 7.17327402680311819721263451580, 8.108936289661534573838202358758, 10.12031867483186501998928159364, 11.74303211748106497918450787313, 12.56249063453334051453101661395, 14.63507563229165500020002978840, 15.66786874144147117978132179447
