| L(s) = 1 | − 2-s − 3-s + 6-s − 4·7-s + 8-s + 3·9-s − 6·13-s + 4·14-s − 16-s − 7·17-s − 3·18-s + 7·19-s + 4·21-s − 2·23-s − 24-s + 6·26-s − 8·27-s − 10·29-s − 4·31-s + 7·34-s + 8·37-s − 7·38-s + 6·39-s − 2·41-s − 4·42-s + 12·43-s + 2·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 9-s − 1.66·13-s + 1.06·14-s − 1/4·16-s − 1.69·17-s − 0.707·18-s + 1.60·19-s + 0.872·21-s − 0.417·23-s − 0.204·24-s + 1.17·26-s − 1.53·27-s − 1.85·29-s − 0.718·31-s + 1.20·34-s + 1.31·37-s − 1.13·38-s + 0.960·39-s − 0.312·41-s − 0.617·42-s + 1.82·43-s + 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1271815942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1271815942\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 15 T + 128 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−9.945375299050809169687558083325, −9.863539739500078601203581498044, −9.458727408915369424747099895819, −9.183689866291951142782107207013, −8.929679286484857229234258393949, −8.012359466433770125134083866667, −7.50708464308209509550715752550, −7.30412621269998237350727960862, −7.15488639386538675401656096160, −6.36835858304964599464030295006, −6.05960659677443724933872505973, −5.62477594214103091033009849738, −4.80830198253135110570329868018, −4.74498306550593141600686354744, −3.76820332631954001470310510809, −3.69883378923026996499869594817, −2.64898730943208923543592352889, −2.25674862786290202247811211318, −1.38495792163468881428220514014, −0.19823096282507986237701611881,
0.19823096282507986237701611881, 1.38495792163468881428220514014, 2.25674862786290202247811211318, 2.64898730943208923543592352889, 3.69883378923026996499869594817, 3.76820332631954001470310510809, 4.74498306550593141600686354744, 4.80830198253135110570329868018, 5.62477594214103091033009849738, 6.05960659677443724933872505973, 6.36835858304964599464030295006, 7.15488639386538675401656096160, 7.30412621269998237350727960862, 7.50708464308209509550715752550, 8.012359466433770125134083866667, 8.929679286484857229234258393949, 9.183689866291951142782107207013, 9.458727408915369424747099895819, 9.863539739500078601203581498044, 9.945375299050809169687558083325
