| L(s) = 1 | − 3·4-s − 4·5-s − 2·7-s + 9-s − 4·13-s + 5·16-s + 12·20-s + 2·25-s + 6·28-s − 2·29-s + 8·35-s − 3·36-s − 4·45-s + 3·49-s + 12·52-s + 12·53-s + 24·59-s − 2·63-s − 3·64-s + 16·65-s + 8·67-s − 20·80-s + 81-s − 24·83-s + 8·91-s − 6·100-s + 16·103-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.10·13-s + 5/4·16-s + 2.68·20-s + 2/5·25-s + 1.13·28-s − 0.371·29-s + 1.35·35-s − 1/2·36-s − 0.596·45-s + 3/7·49-s + 1.66·52-s + 1.64·53-s + 3.12·59-s − 0.251·63-s − 3/8·64-s + 1.98·65-s + 0.977·67-s − 2.23·80-s + 1/9·81-s − 2.63·83-s + 0.838·91-s − 3/5·100-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.548328603682253116112277067930, −8.135249988088522958713590759000, −7.48292463845768060212531982747, −7.25047783802838427330028450541, −6.85540750527117050478666887987, −6.04718499105037929041596410377, −5.33127488666172597415900037702, −5.14083448762531515766594705626, −4.20755242463912897426431158801, −4.13559084050773741974089362056, −3.75007661297492683580937601463, −3.05279040294571468472224368289, −2.22318003493217483206616099006, −0.73756874722956271567483825317, 0,
0.73756874722956271567483825317, 2.22318003493217483206616099006, 3.05279040294571468472224368289, 3.75007661297492683580937601463, 4.13559084050773741974089362056, 4.20755242463912897426431158801, 5.14083448762531515766594705626, 5.33127488666172597415900037702, 6.04718499105037929041596410377, 6.85540750527117050478666887987, 7.25047783802838427330028450541, 7.48292463845768060212531982747, 8.135249988088522958713590759000, 8.548328603682253116112277067930
