| L(s) = 1 | − 1.56e5·5-s − 9.09e5·7-s + 9.56e6·9-s + 3.18e7·11-s − 4.65e7·17-s − 1.78e9·19-s − 1.31e10·23-s + 6.10e9·25-s + 1.42e11·35-s + 2.61e11·43-s − 1.49e12·45-s + 1.84e11·47-s + 6.78e11·49-s − 4.97e12·55-s + 4.29e12·61-s − 8.70e12·63-s − 2.06e13·73-s − 2.89e13·77-s + 6.86e13·81-s + 8.29e13·83-s + 7.27e12·85-s + 2.79e14·95-s + 3.04e14·99-s − 2.29e14·101-s + 2.05e15·115-s + 4.23e13·119-s + 3.79e14·121-s + ⋯ |
| L(s) = 1 | − 1.99·5-s − 1.10·7-s + 2·9-s + 1.63·11-s − 0.113·17-s − 2·19-s − 3.86·23-s + 25-s + 2.20·35-s + 0.962·43-s − 3.99·45-s + 0.364·47-s + 49-s − 3.27·55-s + 1.36·61-s − 2.20·63-s − 1.86·73-s − 1.80·77-s + 3·81-s + 3.05·83-s + 0.226·85-s + 3.99·95-s + 3.27·99-s − 2.13·101-s + 7.72·115-s + 0.125·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+7)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.4996280612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4996280612\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p^{7} T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 156231 T + 18304609736 T^{2} + 156231 p^{14} T^{3} + p^{28} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 909715 T + 149358308376 T^{2} + 909715 p^{14} T^{3} + p^{28} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 31872117 T + 636082008478448 T^{2} - 31872117 p^{14} T^{3} + p^{28} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 46566255 T - 166209410454675904 T^{2} + 46566255 p^{14} T^{3} + p^{28} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6579461310 T + p^{14} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 261715514845 T - \)\(53\!\cdots\!24\)\( T^{2} - 261715514845 p^{14} T^{3} + p^{28} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 184507297725 T - \)\(22\!\cdots\!44\)\( T^{2} - 184507297725 p^{14} T^{3} + p^{28} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 4298781327017 T + \)\(86\!\cdots\!48\)\( T^{2} - 4298781327017 p^{14} T^{3} + p^{28} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 20629557975575 T + \)\(30\!\cdots\!16\)\( T^{2} + 20629557975575 p^{14} T^{3} + p^{28} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 41464886203530 T + p^{14} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{7} T )^{2}( 1 + p^{7} T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.94221781350487643554738478438, −11.81937061631526257066916819001, −10.67170870283068653611861985112, −10.43959905991932362095360747938, −9.592617636316513067228231351826, −9.458020222804770361576100839219, −8.317841984895936983119059121632, −8.144429518491696471331043445619, −7.36428532778927118334141111853, −6.97602042375139053448716244199, −6.29551697219106568499046817652, −6.01131872389653276975097717626, −4.31655194034993127969330661515, −4.28087486208945203585323061539, −3.85171710174896355625707949312, −3.61884825226637346711699572839, −2.18272291789212066860183507734, −1.81538230715193448427952117805, −0.846915401983709476457901551578, −0.20076682175409194607828458257,
0.20076682175409194607828458257, 0.846915401983709476457901551578, 1.81538230715193448427952117805, 2.18272291789212066860183507734, 3.61884825226637346711699572839, 3.85171710174896355625707949312, 4.28087486208945203585323061539, 4.31655194034993127969330661515, 6.01131872389653276975097717626, 6.29551697219106568499046817652, 6.97602042375139053448716244199, 7.36428532778927118334141111853, 8.144429518491696471331043445619, 8.317841984895936983119059121632, 9.458020222804770361576100839219, 9.592617636316513067228231351826, 10.43959905991932362095360747938, 10.67170870283068653611861985112, 11.81937061631526257066916819001, 11.94221781350487643554738478438
