| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s + 4·7-s + 4·8-s + 3·12-s + 2·13-s + 8·14-s + 5·16-s + 4·21-s + 4·24-s − 2·25-s + 4·26-s − 4·27-s + 12·28-s + 6·32-s + 4·37-s + 2·39-s + 18·41-s + 8·42-s + 2·43-s − 6·47-s + 5·48-s − 2·49-s − 4·50-s + 6·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s + 1.51·7-s + 1.41·8-s + 0.866·12-s + 0.554·13-s + 2.13·14-s + 5/4·16-s + 0.872·21-s + 0.816·24-s − 2/5·25-s + 0.784·26-s − 0.769·27-s + 2.26·28-s + 1.06·32-s + 0.657·37-s + 0.320·39-s + 2.81·41-s + 1.23·42-s + 0.304·43-s − 0.875·47-s + 0.721·48-s − 2/7·49-s − 0.565·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1362828 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1362828 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.038293536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.038293536\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 337 | $C_2$ | \( 1 + 14 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.87583153812389529753864342269, −7.55349148307265901417394697337, −7.17589970256136214345232401912, −6.45104134266092276714876790012, −6.10215384253171977615020066510, −5.70790541646475568646379967818, −5.24758755085125743651018842526, −4.71219271239630540409033953310, −4.36700738886294933622429707831, −3.97333899625201333139553152602, −3.41690310785195852844732855476, −2.83791141781274636547714089829, −2.27638073010587782446614450314, −1.77362658157355036409059112134, −1.09404655042232194138291587850,
1.09404655042232194138291587850, 1.77362658157355036409059112134, 2.27638073010587782446614450314, 2.83791141781274636547714089829, 3.41690310785195852844732855476, 3.97333899625201333139553152602, 4.36700738886294933622429707831, 4.71219271239630540409033953310, 5.24758755085125743651018842526, 5.70790541646475568646379967818, 6.10215384253171977615020066510, 6.45104134266092276714876790012, 7.17589970256136214345232401912, 7.55349148307265901417394697337, 7.87583153812389529753864342269
