L(s) = 1 | − 2·3-s − 3·4-s + 2·7-s + 9-s + 6·12-s + 2·13-s + 5·16-s − 8·19-s − 4·21-s − 25-s + 4·27-s − 6·28-s − 3·36-s + 16·37-s − 4·39-s − 16·43-s − 10·48-s − 11·49-s − 6·52-s + 16·57-s − 2·61-s + 2·63-s − 3·64-s − 14·67-s − 22·73-s + 2·75-s + 24·76-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 3/2·4-s + 0.755·7-s + 1/3·9-s + 1.73·12-s + 0.554·13-s + 5/4·16-s − 1.83·19-s − 0.872·21-s − 1/5·25-s + 0.769·27-s − 1.13·28-s − 1/2·36-s + 2.63·37-s − 0.640·39-s − 2.43·43-s − 1.44·48-s − 1.57·49-s − 0.832·52-s + 2.11·57-s − 0.256·61-s + 0.251·63-s − 3/8·64-s − 1.71·67-s − 2.57·73-s + 0.230·75-s + 2.75·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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
−10.14754439690118852610085973818, −9.728214552927016397819296111468, −9.094278745178775977258475921416, −8.461632808149863767883821175577, −8.287570693733701613432166498970, −7.66039763333771123227596439320, −6.60452213843905474925590905017, −6.27968150345344323874847863285, −5.60157612088078313975836309422, −5.04083511992249882455969039728, −4.34489806993819155894459392652, −4.25666915587159511841100490901, −2.98861840356708043816285597500, −1.50537219444186445754147428019, 0,
1.50537219444186445754147428019, 2.98861840356708043816285597500, 4.25666915587159511841100490901, 4.34489806993819155894459392652, 5.04083511992249882455969039728, 5.60157612088078313975836309422, 6.27968150345344323874847863285, 6.60452213843905474925590905017, 7.66039763333771123227596439320, 8.287570693733701613432166498970, 8.461632808149863767883821175577, 9.094278745178775977258475921416, 9.728214552927016397819296111468, 10.14754439690118852610085973818