| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s + 8-s + 6·9-s − 4·11-s − 3·12-s − 2·13-s + 16-s + 6·18-s − 4·22-s − 8·23-s − 3·24-s − 9·25-s − 2·26-s − 9·27-s + 32-s + 12·33-s + 6·36-s + 6·37-s + 6·39-s − 4·44-s − 8·46-s + 26·47-s − 3·48-s − 13·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.353·8-s + 2·9-s − 1.20·11-s − 0.866·12-s − 0.554·13-s + 1/4·16-s + 1.41·18-s − 0.852·22-s − 1.66·23-s − 0.612·24-s − 9/5·25-s − 0.392·26-s − 1.73·27-s + 0.176·32-s + 2.08·33-s + 36-s + 0.986·37-s + 0.960·39-s − 0.603·44-s − 1.17·46-s + 3.79·47-s − 0.433·48-s − 1.85·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.28025762974353066563394220777, −9.614361104441401414050869019861, −8.963928209525859216279788816156, −7.931818475756913627713932784539, −7.45985560667588965978165906144, −7.42085241738992224550819026024, −6.19695969612857050602479487947, −5.88908450915949561478136129218, −5.84229781714116440214520488100, −4.73920502052987077827176335423, −4.61153453491182141362175201622, −3.82010897415061793912288152862, −2.71625515373288996093576035666, −1.77354291627119599659363661895, 0,
1.77354291627119599659363661895, 2.71625515373288996093576035666, 3.82010897415061793912288152862, 4.61153453491182141362175201622, 4.73920502052987077827176335423, 5.84229781714116440214520488100, 5.88908450915949561478136129218, 6.19695969612857050602479487947, 7.42085241738992224550819026024, 7.45985560667588965978165906144, 7.931818475756913627713932784539, 8.963928209525859216279788816156, 9.614361104441401414050869019861, 10.28025762974353066563394220777
