| L(s) = 1 | − 2-s − 4·3-s − 4·5-s + 4·6-s − 3·7-s + 8-s + 6·9-s + 4·10-s − 2·11-s − 4·13-s + 3·14-s + 16·15-s − 16-s − 3·17-s − 6·18-s − 8·19-s + 12·21-s + 2·22-s + 3·23-s − 4·24-s + 6·25-s + 4·26-s + 4·27-s − 2·29-s − 16·30-s + 8·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 1.78·5-s + 1.63·6-s − 1.13·7-s + 0.353·8-s + 2·9-s + 1.26·10-s − 0.603·11-s − 1.10·13-s + 0.801·14-s + 4.13·15-s − 1/4·16-s − 0.727·17-s − 1.41·18-s − 1.83·19-s + 2.61·21-s + 0.426·22-s + 0.625·23-s − 0.816·24-s + 6/5·25-s + 0.784·26-s + 0.769·27-s − 0.371·29-s − 2.92·30-s + 1.39·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11708 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11708 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 2927 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 24 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 17 T + 168 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 157 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 180 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−16.8669332300, −16.6342608365, −16.0468600708, −15.6842679906, −15.2533032206, −14.7622730541, −13.8836512830, −12.9512227987, −12.6530444329, −12.3188614226, −11.7527287151, −11.2993922937, −10.8916938968, −10.5243153604, −9.96772694455, −9.13947946733, −8.51905641654, −7.91228428159, −7.15896578710, −6.71338391687, −6.16833913863, −5.35685469143, −4.66638833459, −4.13531186184, −2.86787183122, 0, 0,
2.86787183122, 4.13531186184, 4.66638833459, 5.35685469143, 6.16833913863, 6.71338391687, 7.15896578710, 7.91228428159, 8.51905641654, 9.13947946733, 9.96772694455, 10.5243153604, 10.8916938968, 11.2993922937, 11.7527287151, 12.3188614226, 12.6530444329, 12.9512227987, 13.8836512830, 14.7622730541, 15.2533032206, 15.6842679906, 16.0468600708, 16.6342608365, 16.8669332300
