| L(s) = 1 | − 2-s − 3·5-s + 8-s + 3·10-s − 3·11-s − 13-s − 16-s + 6·17-s + 14·19-s + 3·22-s − 9·23-s + 5·25-s + 26-s + 3·29-s + 8·31-s − 6·34-s − 2·37-s − 14·38-s − 3·40-s − 3·41-s + 43-s + 9·46-s − 5·50-s − 6·53-s + 9·55-s − 3·58-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s + 0.353·8-s + 0.948·10-s − 0.904·11-s − 0.277·13-s − 1/4·16-s + 1.45·17-s + 3.21·19-s + 0.639·22-s − 1.87·23-s + 25-s + 0.196·26-s + 0.557·29-s + 1.43·31-s − 1.02·34-s − 0.328·37-s − 2.27·38-s − 0.474·40-s − 0.468·41-s + 0.152·43-s + 1.32·46-s − 0.707·50-s − 0.824·53-s + 1.21·55-s − 0.393·58-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6060843821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6060843821\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−9.290981446306554925699982864772, −8.505605718309281142603713440895, −8.061979431981342542534257037414, −7.904790115246416147630769096554, −7.75285502443630811403715359906, −7.38848489857916687051915165467, −7.04013205317823371423477137727, −6.42117742284726696914253610642, −5.99292637265878913364615025477, −5.39887763587182953376114142405, −5.18654243284805141026135550587, −4.85633877957346688108075555313, −4.13832995751934852907245563702, −3.88546006306761298638424433377, −3.30421740220260141441817031195, −2.88750228013920885480686459987, −2.63052674334801285546256597562, −1.42577152465430470761977922312, −1.20977495287210723993184779910, −0.33310718636645553377475364910,
0.33310718636645553377475364910, 1.20977495287210723993184779910, 1.42577152465430470761977922312, 2.63052674334801285546256597562, 2.88750228013920885480686459987, 3.30421740220260141441817031195, 3.88546006306761298638424433377, 4.13832995751934852907245563702, 4.85633877957346688108075555313, 5.18654243284805141026135550587, 5.39887763587182953376114142405, 5.99292637265878913364615025477, 6.42117742284726696914253610642, 7.04013205317823371423477137727, 7.38848489857916687051915165467, 7.75285502443630811403715359906, 7.904790115246416147630769096554, 8.061979431981342542534257037414, 8.505605718309281142603713440895, 9.290981446306554925699982864772
