| L(s) = 1 | − 8.19e3·4-s + 1.53e5·7-s + 5.03e7·16-s + 4.88e8·25-s − 1.25e9·28-s − 5.65e9·37-s + 4.71e8·43-s + 9.72e9·49-s − 2.74e11·64-s − 3.02e11·67-s − 8.88e11·79-s − 4.00e12·100-s − 2.73e12·109-s + 7.72e12·112-s − 6.27e12·121-s + 127-s + 131-s + 137-s + 139-s + 4.63e13·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.17e13·169-s − 3.86e12·172-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.30·7-s + 3·16-s + 2·25-s − 2.60·28-s − 2.20·37-s + 0.0746·43-s + 0.702·49-s − 4·64-s − 3.33·67-s − 3.65·79-s − 4·100-s − 1.62·109-s + 3.91·112-s − 2·121-s + 4.40·148-s + 1.79·169-s − 0.149·172-s + 2.60·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+6)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.8150592924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8150592924\) |
| \(L(7)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 153502 T + p^{12} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{12} T^{2} )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{12} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 9397582 T + p^{12} T^{2} )( 1 + 9397582 T + p^{12} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 17886962 T + p^{12} T^{2} )( 1 + 17886962 T + p^{12} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{12} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{12} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 530187838 T + p^{12} T^{2} )( 1 + 530187838 T + p^{12} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2826257618 T + p^{12} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 235885102 T + p^{12} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{12} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74063873522 T + p^{12} T^{2} )( 1 + 74063873522 T + p^{12} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 151031344462 T + p^{12} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{12} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 104459767778 T + p^{12} T^{2} )( 1 + 104459767778 T + p^{12} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 444304748158 T + p^{12} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{6} T )^{2}( 1 + p^{6} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1662757858942 T + p^{12} T^{2} )( 1 + 1662757858942 T + p^{12} T^{2} ) \) |
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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
−12.88409539685420982749332190880, −12.17237864832360667812177773708, −11.80623028833303781749275615991, −10.70733734335517097324909638121, −10.57149976781089670175780477154, −9.813812053715414809571653491244, −9.076545679577644356849803540237, −8.602052976608002288075494471459, −8.444050262149757929536734447746, −7.58967901732018030179804224423, −6.99748908956586028482645171377, −5.80589015379023807315858082667, −5.32919213527014969704064735981, −4.58032373011455922027165907416, −4.52969854372148154225065310355, −3.53746636720592111360587561375, −2.86967456972757634547441715681, −1.50719270248146623992593010845, −1.23014312571678428756116582996, −0.26704049153602949220158249021,
0.26704049153602949220158249021, 1.23014312571678428756116582996, 1.50719270248146623992593010845, 2.86967456972757634547441715681, 3.53746636720592111360587561375, 4.52969854372148154225065310355, 4.58032373011455922027165907416, 5.32919213527014969704064735981, 5.80589015379023807315858082667, 6.99748908956586028482645171377, 7.58967901732018030179804224423, 8.444050262149757929536734447746, 8.602052976608002288075494471459, 9.076545679577644356849803540237, 9.813812053715414809571653491244, 10.57149976781089670175780477154, 10.70733734335517097324909638121, 11.80623028833303781749275615991, 12.17237864832360667812177773708, 12.88409539685420982749332190880
