| L(s) = 1 | + 2·2-s − 4-s + 5-s + 4·7-s − 8·8-s − 3·9-s + 2·10-s − 5·11-s + 5·13-s + 8·14-s − 7·16-s − 3·17-s − 6·18-s − 19-s − 20-s − 10·22-s − 3·23-s + 5·25-s + 10·26-s − 4·28-s + 29-s + 14·32-s − 6·34-s + 4·35-s + 3·36-s − 3·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 0.447·5-s + 1.51·7-s − 2.82·8-s − 9-s + 0.632·10-s − 1.50·11-s + 1.38·13-s + 2.13·14-s − 7/4·16-s − 0.727·17-s − 1.41·18-s − 0.229·19-s − 0.223·20-s − 2.13·22-s − 0.625·23-s + 25-s + 1.96·26-s − 0.755·28-s + 0.185·29-s + 2.47·32-s − 1.02·34-s + 0.676·35-s + 1/2·36-s − 0.493·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.184264143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184264143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−14.86320811388511255413287212482, −14.66473203835879405522644063139, −13.88680957493127461701048482296, −13.71854804953023013846994232051, −13.31543831428165683678846709223, −12.74061879927651581759917057161, −12.05106951289808834686854428832, −11.57286163977191096296197937749, −10.68987656203190730086364755354, −10.52931791595191573691488646241, −9.183117331261402809403179304908, −8.821719695832177674354867956115, −8.349220728424878624324179693032, −7.70328543557386928749158187665, −6.08576838044891827252090132427, −5.92206473894595284595641012818, −4.92887929223857339931243733265, −4.77534757262665122469953113926, −3.67265667809824048879216192133, −2.62442707553975429084638472217,
2.62442707553975429084638472217, 3.67265667809824048879216192133, 4.77534757262665122469953113926, 4.92887929223857339931243733265, 5.92206473894595284595641012818, 6.08576838044891827252090132427, 7.70328543557386928749158187665, 8.349220728424878624324179693032, 8.821719695832177674354867956115, 9.183117331261402809403179304908, 10.52931791595191573691488646241, 10.68987656203190730086364755354, 11.57286163977191096296197937749, 12.05106951289808834686854428832, 12.74061879927651581759917057161, 13.31543831428165683678846709223, 13.71854804953023013846994232051, 13.88680957493127461701048482296, 14.66473203835879405522644063139, 14.86320811388511255413287212482
