| L(s) = 1 | + 300·5-s − 6.78e3·13-s − 1.03e4·17-s + 3.62e4·25-s + 6.42e4·29-s + 1.52e5·37-s + 1.40e5·41-s + 1.29e5·49-s + 1.33e5·53-s + 5.14e5·61-s − 2.03e6·65-s + 4.86e5·73-s − 3.10e6·85-s + 1.37e6·89-s − 1.88e6·97-s − 5.19e5·101-s + 2.04e6·109-s + 2.62e6·113-s + 1.36e6·121-s − 5.62e5·125-s + 127-s + 131-s + 137-s + 139-s + 1.92e7·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 12/5·5-s − 3.08·13-s − 2.10·17-s + 2.31·25-s + 2.63·29-s + 3.00·37-s + 2.03·41-s + 1.09·49-s + 0.899·53-s + 2.26·61-s − 7.41·65-s + 1.25·73-s − 5.05·85-s + 1.94·89-s − 2.06·97-s − 0.504·101-s + 1.58·109-s + 1.82·113-s + 0.770·121-s − 0.287·125-s + 6.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(6.247507124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.247507124\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 6 p^{2} T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 129266 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1365410 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3394 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 5178 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 47709890 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 280146530 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 32142 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 711526610 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 76150 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 70038 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2451652130 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 1425021694 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 66942 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 68178858910 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 257014 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 77748332930 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138474492194 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 243442 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 260585085842 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 415389237694 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 686766 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 942686 T + p^{6} T^{2} )^{2} \) |
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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
−9.756143099232904038835634405941, −9.699206310312513556753552735171, −9.277282245757247441273051936473, −8.960215371059224530684248281559, −8.198191695899530012228108148979, −7.81886044606999150861829131126, −6.99946177092355853729827341407, −6.92268561153607675644115399436, −6.32791024213688181320713784821, −5.90558310753398397622519386148, −5.48728840372877129110644576804, −4.89245906020186275117435719437, −4.50590655884965072749849327006, −4.23074939844459689125562052727, −2.75304697716430135239551908850, −2.57855636225430603055137118011, −2.24058881168540312572072220342, −1.99307725054413824151272828162, −0.73602712848612035523399537833, −0.67074687321700609260218292111,
0.67074687321700609260218292111, 0.73602712848612035523399537833, 1.99307725054413824151272828162, 2.24058881168540312572072220342, 2.57855636225430603055137118011, 2.75304697716430135239551908850, 4.23074939844459689125562052727, 4.50590655884965072749849327006, 4.89245906020186275117435719437, 5.48728840372877129110644576804, 5.90558310753398397622519386148, 6.32791024213688181320713784821, 6.92268561153607675644115399436, 6.99946177092355853729827341407, 7.81886044606999150861829131126, 8.198191695899530012228108148979, 8.960215371059224530684248281559, 9.277282245757247441273051936473, 9.699206310312513556753552735171, 9.756143099232904038835634405941
