| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s − 3·9-s + 2·11-s − 4·15-s − 4·17-s − 6·19-s + 4·21-s + 2·23-s + 3·25-s − 14·27-s + 6·29-s − 4·31-s + 4·33-s − 4·35-s + 14·37-s − 2·41-s + 2·43-s + 6·45-s + 6·47-s − 4·49-s − 8·51-s − 4·55-s − 12·57-s + 22·61-s − 6·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s − 9-s + 0.603·11-s − 1.03·15-s − 0.970·17-s − 1.37·19-s + 0.872·21-s + 0.417·23-s + 3/5·25-s − 2.69·27-s + 1.11·29-s − 0.718·31-s + 0.696·33-s − 0.676·35-s + 2.30·37-s − 0.312·41-s + 0.304·43-s + 0.894·45-s + 0.875·47-s − 4/7·49-s − 1.12·51-s − 0.539·55-s − 1.58·57-s + 2.81·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.725703144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.725703144\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 116 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 22 T + 236 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 151 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 143 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 152 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 172 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.054866855554606297849069495863, −7.998155238728766890973716181497, −7.41430840299340308158126370484, −7.25761466847876599140018357089, −6.58485270570982091017455445785, −6.50509056478020244294582560548, −6.00674917712227301266124378264, −5.71137200873429198312533536763, −5.10836042855418741933892794581, −4.86487080827225321194029574494, −4.35981605019377907361650045009, −4.11708009168489018025154072564, −3.68194673035235556053046766036, −3.45852364120179009544075224839, −2.72966359966700078912483217045, −2.58619303707378642593107364519, −2.21172211855283273085616837390, −1.74573970475029515510322772993, −0.927340122161079986807736953059, −0.42095376940755160805644709412,
0.42095376940755160805644709412, 0.927340122161079986807736953059, 1.74573970475029515510322772993, 2.21172211855283273085616837390, 2.58619303707378642593107364519, 2.72966359966700078912483217045, 3.45852364120179009544075224839, 3.68194673035235556053046766036, 4.11708009168489018025154072564, 4.35981605019377907361650045009, 4.86487080827225321194029574494, 5.10836042855418741933892794581, 5.71137200873429198312533536763, 6.00674917712227301266124378264, 6.50509056478020244294582560548, 6.58485270570982091017455445785, 7.25761466847876599140018357089, 7.41430840299340308158126370484, 7.998155238728766890973716181497, 8.054866855554606297849069495863
