| L(s) = 1 | + 486·9-s + 3.89e3·17-s + 4.09e4·25-s + 1.33e5·41-s + 4.47e5·49-s − 4.54e5·73-s + 1.77e5·81-s − 1.85e6·89-s + 2.06e5·97-s − 3.21e5·113-s − 6.77e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.89e6·153-s + 157-s + 163-s + 167-s + 9.59e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 0.792·17-s + 2.61·25-s + 1.93·41-s + 3.80·49-s − 1.16·73-s + 1/3·81-s − 2.63·89-s + 0.226·97-s − 0.222·113-s − 3.82·121-s + 0.528·153-s + 1.98·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6697075335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6697075335\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 818 p^{2} T^{2} + p^{12} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 223634 T^{2} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 3387170 T^{2} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4799666 T^{2} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 974 T + p^{6} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 93137474 T^{2} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 142914526 T^{2} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1099423874 T^{2} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1529771762 T^{2} + p^{12} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2483043986 T^{2} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 33298 T + p^{6} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 12375158690 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 16196676482 T^{2} + p^{12} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 17210384930 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 78728080610 T^{2} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 103017496754 T^{2} + p^{12} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 112517388290 T^{2} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 228753356258 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 113618 T + p^{6} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 52868683442 T^{2} + p^{12} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 321573289538 T^{2} + p^{12} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 464290 T + p^{6} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 51694 T + p^{6} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.13068289212950544989648481682, −6.88683270479112776284097158951, −6.69566023616122601877477426754, −6.64214597052214642252139018581, −6.00115111000239269471035727375, −5.97309927041733208358818128384, −5.61667677396050749782403784789, −5.36610243705979988651535816864, −5.17998987114103584993722367468, −4.91766154155027062355626862646, −4.35424358274137547603134643390, −4.25853908931475667710260078176, −4.21564653161349536950413878079, −3.78318509909862645926873914102, −3.40097561072295266849914413708, −3.08380121660886981772636107745, −2.78159227170254103907430742774, −2.52317661262006623671400804456, −2.34977284611046894645613867236, −1.89545371936674483608457847456, −1.25188554242355120082954102966, −1.23505053167322734640478909703, −0.936839107106656775970695514251, −0.74780355906084811322795620247, −0.07645677380433475183506644097,
0.07645677380433475183506644097, 0.74780355906084811322795620247, 0.936839107106656775970695514251, 1.23505053167322734640478909703, 1.25188554242355120082954102966, 1.89545371936674483608457847456, 2.34977284611046894645613867236, 2.52317661262006623671400804456, 2.78159227170254103907430742774, 3.08380121660886981772636107745, 3.40097561072295266849914413708, 3.78318509909862645926873914102, 4.21564653161349536950413878079, 4.25853908931475667710260078176, 4.35424358274137547603134643390, 4.91766154155027062355626862646, 5.17998987114103584993722367468, 5.36610243705979988651535816864, 5.61667677396050749782403784789, 5.97309927041733208358818128384, 6.00115111000239269471035727375, 6.64214597052214642252139018581, 6.69566023616122601877477426754, 6.88683270479112776284097158951, 7.13068289212950544989648481682
