| L(s) = 1 | + 5·9-s + 86·11-s − 70·19-s − 320·29-s − 84·31-s − 406·41-s + 650·49-s − 560·59-s − 1.03e3·61-s − 824·71-s + 1.02e3·79-s − 704·81-s + 1.89e3·89-s + 430·99-s + 2.60e3·101-s − 2.14e3·109-s + 2.88e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.61e3·169-s + ⋯ |
| L(s) = 1 | + 5/27·9-s + 2.35·11-s − 0.845·19-s − 2.04·29-s − 0.486·31-s − 1.54·41-s + 1.89·49-s − 1.23·59-s − 2.17·61-s − 1.37·71-s + 1.45·79-s − 0.965·81-s + 2.25·89-s + 0.436·99-s + 2.56·101-s − 1.88·109-s + 2.16·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.64·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.392568432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.392568432\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 650 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 43 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3610 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1545 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 35 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1910 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 160 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2710 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 203 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 150550 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 169230 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291030 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 280 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 518 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 581645 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 412 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 195865 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 510 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 539845 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 945 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 272830 T^{2} + p^{6} T^{4} \) |
| 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
−11.04800686206738586350046521188, −10.58124976652088098601141289543, −10.33878462683401464361002083989, −9.367365767009723180605233287437, −9.356631225121501439067614207365, −8.916328029373309850884494876079, −8.524008431174238540375117713795, −7.55290888388414223994073033392, −7.52331073564011179322902942975, −6.73173794090874217727267818602, −6.38556181494375497412423532020, −5.96207732822383775842538285837, −5.30984016629710464406361477787, −4.58604461128653227671891498493, −4.00718254765487309133302652114, −3.72714629973999664474696684080, −2.99649552687594516601490132911, −1.77430323532526187344602828642, −1.67333006552176801852205846904, −0.52430139752224847275078268789,
0.52430139752224847275078268789, 1.67333006552176801852205846904, 1.77430323532526187344602828642, 2.99649552687594516601490132911, 3.72714629973999664474696684080, 4.00718254765487309133302652114, 4.58604461128653227671891498493, 5.30984016629710464406361477787, 5.96207732822383775842538285837, 6.38556181494375497412423532020, 6.73173794090874217727267818602, 7.52331073564011179322902942975, 7.55290888388414223994073033392, 8.524008431174238540375117713795, 8.916328029373309850884494876079, 9.356631225121501439067614207365, 9.367365767009723180605233287437, 10.33878462683401464361002083989, 10.58124976652088098601141289543, 11.04800686206738586350046521188
