| L(s) = 1 | − 2-s + 8-s − 8·13-s − 16-s + 6·17-s − 2·19-s + 5·25-s + 8·26-s + 12·29-s + 4·31-s − 6·34-s − 2·37-s + 2·38-s − 12·41-s + 16·43-s − 12·47-s − 5·50-s + 6·53-s − 12·58-s − 6·59-s − 8·61-s − 4·62-s + 64-s + 4·67-s − 2·73-s + 2·74-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.353·8-s − 2.21·13-s − 1/4·16-s + 1.45·17-s − 0.458·19-s + 25-s + 1.56·26-s + 2.22·29-s + 0.718·31-s − 1.02·34-s − 0.328·37-s + 0.324·38-s − 1.87·41-s + 2.43·43-s − 1.75·47-s − 0.707·50-s + 0.824·53-s − 1.57·58-s − 0.781·59-s − 1.02·61-s − 0.508·62-s + 1/8·64-s + 0.488·67-s − 0.234·73-s + 0.232·74-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003958300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003958300\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.24050168938634619756520511423, −9.947962399167162574763516562225, −9.562839283642452873150273824419, −9.202208327398030096372454610694, −8.579070835181114491590579675972, −8.290591895518752341408238230342, −7.893845413274031277357094778620, −7.43214040408875706848220213907, −7.04484775800697037631545065049, −6.61190330864623345980003208225, −6.16705628437093657687964839083, −5.25382606869139825437544321754, −5.22717740194178482115636502177, −4.48973455666388034502967389192, −4.30707667531637213604764354311, −3.13792778408072637693602168647, −2.96914043439670374576015188211, −2.27167569372610041124362855296, −1.41601001243632920313198798025, −0.57480985419816511141810679089,
0.57480985419816511141810679089, 1.41601001243632920313198798025, 2.27167569372610041124362855296, 2.96914043439670374576015188211, 3.13792778408072637693602168647, 4.30707667531637213604764354311, 4.48973455666388034502967389192, 5.22717740194178482115636502177, 5.25382606869139825437544321754, 6.16705628437093657687964839083, 6.61190330864623345980003208225, 7.04484775800697037631545065049, 7.43214040408875706848220213907, 7.893845413274031277357094778620, 8.290591895518752341408238230342, 8.579070835181114491590579675972, 9.202208327398030096372454610694, 9.562839283642452873150273824419, 9.947962399167162574763516562225, 10.24050168938634619756520511423
