| L(s) = 1 | + 5·2-s − 9·4-s − 80·7-s − 55·8-s − 800·11-s + 120·13-s − 400·14-s − 193·16-s + 1.94e3·17-s − 1.51e3·19-s − 4.00e3·22-s − 1.32e3·23-s + 600·26-s + 720·28-s − 1.30e3·29-s − 5.82e3·31-s − 5.65e3·32-s + 9.70e3·34-s + 1.25e4·37-s − 7.56e3·38-s − 400·41-s − 2.56e4·43-s + 7.20e3·44-s − 6.60e3·46-s + 1.89e4·47-s − 1.43e4·49-s − 1.08e3·52-s + ⋯ |
| L(s) = 1 | + 0.883·2-s − 0.281·4-s − 0.617·7-s − 0.303·8-s − 1.99·11-s + 0.196·13-s − 0.545·14-s − 0.188·16-s + 1.62·17-s − 0.960·19-s − 1.76·22-s − 0.520·23-s + 0.174·26-s + 0.173·28-s − 0.287·29-s − 1.08·31-s − 0.976·32-s + 1.43·34-s + 1.50·37-s − 0.849·38-s − 0.0371·41-s − 2.11·43-s + 0.560·44-s − 0.459·46-s + 1.24·47-s − 0.851·49-s − 0.0553·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 5 T + 17 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 80 T + 20714 T^{2} + 80 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 800 T + 467602 T^{2} + 800 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 120 T + 514186 T^{2} - 120 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1940 T + 3743494 T^{2} - 1940 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1512 T + 1811734 T^{2} + 1512 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1320 T + 4100206 T^{2} + 1320 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1300 T - 17947202 T^{2} + 1300 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5824 T + 58720046 T^{2} + 5824 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12560 T + 102958314 T^{2} - 12560 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 400 T + 164704402 T^{2} + 400 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 25680 T + 399490486 T^{2} + 25680 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18920 T + 546954334 T^{2} - 18920 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 49460 T + 1434563566 T^{2} - 49460 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 63200 T + 2393594098 T^{2} + 63200 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 49116 T + 1219519966 T^{2} + 49116 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6080 T + 319369814 T^{2} + 6080 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 65200 T + 4591816702 T^{2} + 65200 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 97740 T + 5753972086 T^{2} + 97740 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 46288 T + 6429395534 T^{2} + 46288 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 57360 T + 4528611766 T^{2} + 57360 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 87000 T + 10502568898 T^{2} - 87000 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10180 T + 16899916614 T^{2} + 10180 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.13764399397377995224614066235, −10.54138582761428206635334811570, −10.14324719965026221527720846814, −10.00263487673777013475350690389, −8.964894260383938287095586866957, −8.864863842424369576015871612326, −7.966548461927288362166301495835, −7.57952566306347684415192736377, −7.26521083420736437653496386604, −6.17445127414103162941265437063, −5.87045091493323521346934829111, −5.37321906525100004509934674565, −4.75259243107793026029078244010, −4.29455438392342282520932618417, −3.48414062383931890460941667296, −3.05523824434054866961339518125, −2.29749144766252873352355448319, −1.38038064097969684262948450517, 0, 0,
1.38038064097969684262948450517, 2.29749144766252873352355448319, 3.05523824434054866961339518125, 3.48414062383931890460941667296, 4.29455438392342282520932618417, 4.75259243107793026029078244010, 5.37321906525100004509934674565, 5.87045091493323521346934829111, 6.17445127414103162941265437063, 7.26521083420736437653496386604, 7.57952566306347684415192736377, 7.966548461927288362166301495835, 8.864863842424369576015871612326, 8.964894260383938287095586866957, 10.00263487673777013475350690389, 10.14324719965026221527720846814, 10.54138582761428206635334811570, 11.13764399397377995224614066235
