| L(s) = 1 | + 4·2-s + 3-s + 8·4-s + 4·6-s − 10·7-s + 8·8-s + 9-s + 8·12-s − 40·14-s − 4·16-s + 4·18-s − 19-s − 10·21-s + 8·24-s − 25-s + 27-s − 80·28-s + 4·29-s − 32·32-s + 8·36-s − 4·38-s − 40·42-s − 2·43-s − 4·48-s + 61·49-s − 4·50-s − 20·53-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.577·3-s + 4·4-s + 1.63·6-s − 3.77·7-s + 2.82·8-s + 1/3·9-s + 2.30·12-s − 10.6·14-s − 16-s + 0.942·18-s − 0.229·19-s − 2.18·21-s + 1.63·24-s − 1/5·25-s + 0.192·27-s − 15.1·28-s + 0.742·29-s − 5.65·32-s + 4/3·36-s − 0.648·38-s − 6.17·42-s − 0.304·43-s − 0.577·48-s + 61/7·49-s − 0.565·50-s − 2.74·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.028755790922139382896168795846, −8.598738845501152801902892747277, −7.58489736631665346168199395971, −6.85073886350288365363080648917, −6.70211043109012358161209293170, −6.16139530761373947123603078754, −6.12241751609850681792173625996, −5.34668146252658900228619106104, −4.78903745519092577014884991606, −3.96503328962323463176155683264, −3.67840796781373175574076394185, −3.37953587462828366448462440771, −2.64121548040677802901277139388, −2.61687998381550726785939862479, 0,
2.61687998381550726785939862479, 2.64121548040677802901277139388, 3.37953587462828366448462440771, 3.67840796781373175574076394185, 3.96503328962323463176155683264, 4.78903745519092577014884991606, 5.34668146252658900228619106104, 6.12241751609850681792173625996, 6.16139530761373947123603078754, 6.70211043109012358161209293170, 6.85073886350288365363080648917, 7.58489736631665346168199395971, 8.598738845501152801902892747277, 9.028755790922139382896168795846
