| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s + 9-s − 2·10-s + 16-s − 6·17-s + 18-s − 2·20-s − 7·25-s + 2·29-s + 32-s − 6·34-s + 36-s + 6·37-s − 2·40-s − 14·41-s − 2·45-s − 13·49-s − 7·50-s − 4·53-s + 2·58-s + 12·61-s + 64-s − 6·68-s + 72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.447·20-s − 7/5·25-s + 0.371·29-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.986·37-s − 0.316·40-s − 2.18·41-s − 0.298·45-s − 1.85·49-s − 0.989·50-s − 0.549·53-s + 0.262·58-s + 1.53·61-s + 1/8·64-s − 0.727·68-s + 0.117·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242208 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242208 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−8.419032137688102627067453548578, −8.394490307116038147042717026528, −7.63591160976614872787897721858, −7.37146406868997028644156113495, −6.72637701647606885979113354058, −6.34044350417483302017587248899, −5.92279523936627297859457876342, −4.91978241839509558898126368942, −4.88979839840342254384190809437, −4.05799789384584110155782853407, −3.79639747241266837059275722803, −3.08983291588101772491182135602, −2.30356803383598646559796068707, −1.57931178975143369874205650295, 0,
1.57931178975143369874205650295, 2.30356803383598646559796068707, 3.08983291588101772491182135602, 3.79639747241266837059275722803, 4.05799789384584110155782853407, 4.88979839840342254384190809437, 4.91978241839509558898126368942, 5.92279523936627297859457876342, 6.34044350417483302017587248899, 6.72637701647606885979113354058, 7.37146406868997028644156113495, 7.63591160976614872787897721858, 8.394490307116038147042717026528, 8.419032137688102627067453548578
