| L(s) = 1 | − 16·4-s + 192·16-s + 472·25-s − 1.50e3·37-s + 686·49-s − 2.04e3·64-s − 7.55e3·100-s + 9.08e3·109-s + 1.48e3·121-s + 127-s + 131-s + 137-s + 139-s + 2.40e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 1.09e4·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s + 3.77·25-s − 6.68·37-s + 2·49-s − 4·64-s − 7.55·100-s + 7.97·109-s + 1.11·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 13.3·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s − 4·196-s + 0.000361·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.138959938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138959938\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 236 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 740 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 9340 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 9830 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 6284 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 19150 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 376 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 28492 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 512300 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 1001212 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
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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
−8.639732762113364179478430212485, −8.205480006563987035040820337681, −7.929308167024488456621050903713, −7.36052893459965553909994650489, −7.26200532661797214383933458606, −7.04623624308274785183467293402, −6.89711445878708802742566538418, −6.38827593642248416515046915610, −6.21373538213661619310916018758, −5.56187224883309104768616756545, −5.42034267071324018694037001188, −5.37217377469908732004529788107, −4.85943226521527646953576329997, −4.73489512190807667353403942285, −4.59283337708448629931303606356, −4.04269047118428318380418548279, −3.65146401890676786754895521764, −3.41617815242302051941430525025, −3.05239868662989753285474638581, −3.04047188582920982902370406305, −1.96565113700764387311927429282, −1.84415846645721254105135585550, −1.17917962662012441699196501148, −0.57690349490050162402160031150, −0.48709511171951585753736298159,
0.48709511171951585753736298159, 0.57690349490050162402160031150, 1.17917962662012441699196501148, 1.84415846645721254105135585550, 1.96565113700764387311927429282, 3.04047188582920982902370406305, 3.05239868662989753285474638581, 3.41617815242302051941430525025, 3.65146401890676786754895521764, 4.04269047118428318380418548279, 4.59283337708448629931303606356, 4.73489512190807667353403942285, 4.85943226521527646953576329997, 5.37217377469908732004529788107, 5.42034267071324018694037001188, 5.56187224883309104768616756545, 6.21373538213661619310916018758, 6.38827593642248416515046915610, 6.89711445878708802742566538418, 7.04623624308274785183467293402, 7.26200532661797214383933458606, 7.36052893459965553909994650489, 7.929308167024488456621050903713, 8.205480006563987035040820337681, 8.639732762113364179478430212485
