| L(s) = 1 | − 108·9-s + 500·25-s − 1.24e3·53-s + 3.52e3·61-s + 7.29e3·81-s + 5.99e3·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 5.40e4·225-s + 227-s + ⋯ |
| L(s) = 1 | − 4·9-s + 4·25-s − 3.21·53-s + 7.40·61-s + 10·81-s + 5.90·101-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s − 16·225-s + 0.000292·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.667044665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.667044665\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 38 T + 722 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} )( 1 + 38 T + 722 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 90 T + 4050 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} )( 1 + 90 T + 4050 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 + 9774 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 70 T + 2450 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} )( 1 + 70 T + 2450 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 14922 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 70 T + 2450 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} )( 1 + 70 T + 2450 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 378 T + 71442 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} )( 1 + 378 T + 71442 T^{2} + 378 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 + 310 T + p^{3} T^{2} )^{4} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 350 T + 61250 T^{2} - 350 p^{3} T^{3} + p^{6} T^{4} )( 1 + 350 T + 61250 T^{2} + 350 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - 882 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 902 T + 406802 T^{2} - 902 p^{3} T^{3} + p^{6} T^{4} )( 1 + 902 T + 406802 T^{2} + 902 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 71 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1630 T + 1328450 T^{2} - 1630 p^{3} T^{3} + p^{6} T^{4} )( 1 + 1630 T + 1328450 T^{2} + 1630 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2114 T + 2234498 T^{2} - 2114 p^{3} T^{3} + p^{6} T^{4} )( 1 + 2114 T + 2234498 T^{2} + 2114 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 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
−6.74427609543950363013272719585, −6.66741823588049881352802126708, −6.54353147828337698666703729377, −6.19836069450057957923683370302, −6.00266617180120222714258715862, −5.71288533483853666480131884160, −5.53095773382860514302446923384, −5.22198392408249587632132871924, −5.14720708482754466018344338874, −4.89866130620108967124075190529, −4.71194650319374922516667788164, −4.42718308063220226979175489295, −3.83455120889396641225603986057, −3.63177432487925349151270965328, −3.25502484387917690549498134236, −3.16890763955198625754088585273, −3.10794551535238449424817721915, −2.55723349344410007001382285977, −2.37351055017806941999448819779, −2.28997265261417135209137563614, −1.82872458409628457585154922889, −1.13903369046564440223613104004, −0.68327313731946522732718797010, −0.66981311173468473817263537004, −0.34912008491548672885151133734,
0.34912008491548672885151133734, 0.66981311173468473817263537004, 0.68327313731946522732718797010, 1.13903369046564440223613104004, 1.82872458409628457585154922889, 2.28997265261417135209137563614, 2.37351055017806941999448819779, 2.55723349344410007001382285977, 3.10794551535238449424817721915, 3.16890763955198625754088585273, 3.25502484387917690549498134236, 3.63177432487925349151270965328, 3.83455120889396641225603986057, 4.42718308063220226979175489295, 4.71194650319374922516667788164, 4.89866130620108967124075190529, 5.14720708482754466018344338874, 5.22198392408249587632132871924, 5.53095773382860514302446923384, 5.71288533483853666480131884160, 6.00266617180120222714258715862, 6.19836069450057957923683370302, 6.54353147828337698666703729377, 6.66741823588049881352802126708, 6.74427609543950363013272719585
