| L(s) = 1 | − 2·3-s + 4-s + 5-s + 2·7-s + 3·9-s − 2·12-s + 2·13-s − 2·15-s − 3·16-s + 3·17-s − 19-s + 20-s − 4·21-s + 11·23-s − 8·25-s − 4·27-s + 2·28-s − 12·29-s + 5·31-s + 2·35-s + 3·36-s + 7·37-s − 4·39-s + 17·41-s + 6·43-s + 3·45-s + 6·48-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 9-s − 0.577·12-s + 0.554·13-s − 0.516·15-s − 3/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s − 0.872·21-s + 2.29·23-s − 8/5·25-s − 0.769·27-s + 0.377·28-s − 2.22·29-s + 0.898·31-s + 0.338·35-s + 1/2·36-s + 1.15·37-s − 0.640·39-s + 2.65·41-s + 0.914·43-s + 0.447·45-s + 0.866·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.810888613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.810888613\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 11 T + 75 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 85 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 197 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.999355771941668545261327977194, −8.935277039265100822582943894821, −8.277143023806921464707774533334, −7.70203729227585032319553505324, −7.48806138108854568756845997161, −7.28552522292009437632091066648, −6.76657110710362350148903189951, −6.14791439071141835084673715635, −6.08931705665007578295238485036, −5.66853214167679404956862056357, −5.12691903750490668227044250446, −5.07266603886371937056629384492, −4.23456922004618191059849945511, −4.03609835182717613984766672393, −3.60579504317032388492675763627, −2.58899977140774114040893153570, −2.45242438535002073366048705990, −1.79557708431389727238868485879, −1.09521240941419714104075119347, −0.70575579302543209014793159266,
0.70575579302543209014793159266, 1.09521240941419714104075119347, 1.79557708431389727238868485879, 2.45242438535002073366048705990, 2.58899977140774114040893153570, 3.60579504317032388492675763627, 4.03609835182717613984766672393, 4.23456922004618191059849945511, 5.07266603886371937056629384492, 5.12691903750490668227044250446, 5.66853214167679404956862056357, 6.08931705665007578295238485036, 6.14791439071141835084673715635, 6.76657110710362350148903189951, 7.28552522292009437632091066648, 7.48806138108854568756845997161, 7.70203729227585032319553505324, 8.277143023806921464707774533334, 8.935277039265100822582943894821, 8.999355771941668545261327977194
