| L(s) = 1 | + 4-s + 2·7-s + 2·11-s − 3·16-s − 3·17-s + 6·25-s + 2·28-s + 10·29-s + 2·37-s + 4·43-s + 2·44-s − 10·49-s + 4·53-s + 6·59-s − 7·64-s − 3·68-s + 4·77-s − 9·81-s − 12·89-s − 2·97-s + 6·100-s + 10·107-s − 6·112-s − 10·113-s + 10·116-s − 6·119-s + 6·121-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 0.603·11-s − 3/4·16-s − 0.727·17-s + 6/5·25-s + 0.377·28-s + 1.85·29-s + 0.328·37-s + 0.609·43-s + 0.301·44-s − 1.42·49-s + 0.549·53-s + 0.781·59-s − 7/8·64-s − 0.363·68-s + 0.455·77-s − 81-s − 1.27·89-s − 0.203·97-s + 3/5·100-s + 0.966·107-s − 0.566·112-s − 0.940·113-s + 0.928·116-s − 0.550·119-s + 6/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47753 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47753 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680346598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680346598\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 132 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.20031176998892011704503486271, −9.628381701329349363614943833849, −9.035598713483186025412233495073, −8.482936895576042873208258119215, −8.291486593957400338561786433882, −7.37702429122021546111228115401, −6.96661898125283035040685418532, −6.48096804345269936261233278524, −5.98795864547802251275846896815, −5.00869791480462797115881239936, −4.65686567470925407206120683820, −4.01792877493683480786978440635, −2.97069046567786444971729335240, −2.32576461206720290241880861530, −1.27908973548718496722710023918,
1.27908973548718496722710023918, 2.32576461206720290241880861530, 2.97069046567786444971729335240, 4.01792877493683480786978440635, 4.65686567470925407206120683820, 5.00869791480462797115881239936, 5.98795864547802251275846896815, 6.48096804345269936261233278524, 6.96661898125283035040685418532, 7.37702429122021546111228115401, 8.291486593957400338561786433882, 8.482936895576042873208258119215, 9.035598713483186025412233495073, 9.628381701329349363614943833849, 10.20031176998892011704503486271
