| L(s) = 1 | − 8·2-s + 6·3-s + 48·4-s − 50·5-s − 48·6-s − 164·7-s − 256·8-s − 171·9-s + 400·10-s + 488·11-s + 288·12-s + 738·13-s + 1.31e3·14-s − 300·15-s + 1.28e3·16-s + 1.11e3·17-s + 1.36e3·18-s − 1.58e3·19-s − 2.40e3·20-s − 984·21-s − 3.90e3·22-s − 1.05e3·23-s − 1.53e3·24-s + 1.87e3·25-s − 5.90e3·26-s − 810·27-s − 7.87e3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.384·3-s + 3/2·4-s − 0.894·5-s − 0.544·6-s − 1.26·7-s − 1.41·8-s − 0.703·9-s + 1.26·10-s + 1.21·11-s + 0.577·12-s + 1.21·13-s + 1.78·14-s − 0.344·15-s + 5/4·16-s + 0.933·17-s + 0.995·18-s − 1.00·19-s − 1.34·20-s − 0.486·21-s − 1.71·22-s − 0.417·23-s − 0.544·24-s + 3/5·25-s − 1.71·26-s − 0.213·27-s − 1.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 p T + 23 p^{2} T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 164 T + 32650 T^{2} + 164 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 488 T + 365438 T^{2} - 488 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 738 T + 848979 T^{2} - 738 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1112 T + 2809362 T^{2} - 1112 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1584 T + 5433662 T^{2} + 1584 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2158 T + 7774731 T^{2} + 2158 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3418 T + 8850271 T^{2} - 3418 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 152 T + 138691378 T^{2} + 152 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 31482 T + 468948251 T^{2} + 31482 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 468 T + 293708690 T^{2} - 468 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10062 T + 425680975 T^{2} - 10062 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3380 T + 580056174 T^{2} - 3380 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 42512 T + 1054234934 T^{2} + 42512 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 44472 T + 1953881250 T^{2} + 44472 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 119392 T + 6227533342 T^{2} + 119392 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 36126 T + 3886041959 T^{2} + 36126 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 39514 T + 3590009203 T^{2} + 39514 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6768 T + 2139648342 T^{2} - 6768 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 124304 T + 10953383718 T^{2} - 124304 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4356 T + 10727769134 T^{2} + 4356 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 139980 T + 21938866206 T^{2} - 139980 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77188016455480508479953499653, −10.64579251252764085697394731492, −10.12309072727087244221307795455, −9.407032442451427550477939637839, −9.072359548340931631075158935803, −8.772327122756738195917217876453, −8.074485918229511028881191444549, −7.998557215084975424003556943349, −7.05970405197129630782624399909, −6.72194266035848285523163260554, −6.06494053491146712484074291525, −5.91269290675175938903023888216, −4.59843792921716560101932775304, −3.81374994414564632308215860838, −3.20437175340088142005339780921, −3.02756611397653343782089608469, −1.72659722526996983920231035493, −1.20825660732563001188048391456, 0, 0,
1.20825660732563001188048391456, 1.72659722526996983920231035493, 3.02756611397653343782089608469, 3.20437175340088142005339780921, 3.81374994414564632308215860838, 4.59843792921716560101932775304, 5.91269290675175938903023888216, 6.06494053491146712484074291525, 6.72194266035848285523163260554, 7.05970405197129630782624399909, 7.998557215084975424003556943349, 8.074485918229511028881191444549, 8.772327122756738195917217876453, 9.072359548340931631075158935803, 9.407032442451427550477939637839, 10.12309072727087244221307795455, 10.64579251252764085697394731492, 10.77188016455480508479953499653
