| L(s) = 1 | − 96·11-s − 108·17-s − 8·19-s − 238·25-s − 588·41-s + 376·43-s − 98·49-s − 504·59-s + 1.25e3·67-s + 2.01e3·73-s + 1.44e3·83-s − 2.96e3·89-s + 3.64e3·97-s + 2.37e3·107-s + 780·113-s + 4.25e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.66e3·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2.63·11-s − 1.54·17-s − 0.0965·19-s − 1.90·25-s − 2.23·41-s + 1.33·43-s − 2/7·49-s − 1.11·59-s + 2.29·67-s + 3.22·73-s + 1.90·83-s − 3.53·89-s + 3.81·97-s + 2.14·107-s + 0.649·113-s + 3.19·121-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 − 1.21·169-s + 0.000439·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.308545049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308545049\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 238 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2666 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 5666 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 22270 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 56114 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4726 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 294 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 188 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 48146 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 256946 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 252 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 445850 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 628 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 715774 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1006 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 811150 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 1482 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1822 T + p^{3} T^{2} )^{2} \) |
| 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
−8.655430955944584362729689025672, −8.438887117514914577726216529966, −7.926218367997248865946977698429, −7.919340838308036289195149600747, −7.33860569110996719232652059100, −7.01805705152733811111974488598, −6.41693681530330771086557399053, −6.21198395836220509596138147854, −5.53526900193583728001193936892, −5.38537309416874898751543371524, −4.76970075411715799693438002826, −4.70375282267492794202573003259, −3.95567177885665624344381629516, −3.51566297451460690966381109263, −3.06549719186210529068662925749, −2.38479635658726799049317114254, −2.12450256821341808860276300417, −1.84857299912999640188911938285, −0.61188199318278778601744147913, −0.34231743596383099664392146484,
0.34231743596383099664392146484, 0.61188199318278778601744147913, 1.84857299912999640188911938285, 2.12450256821341808860276300417, 2.38479635658726799049317114254, 3.06549719186210529068662925749, 3.51566297451460690966381109263, 3.95567177885665624344381629516, 4.70375282267492794202573003259, 4.76970075411715799693438002826, 5.38537309416874898751543371524, 5.53526900193583728001193936892, 6.21198395836220509596138147854, 6.41693681530330771086557399053, 7.01805705152733811111974488598, 7.33860569110996719232652059100, 7.919340838308036289195149600747, 7.926218367997248865946977698429, 8.438887117514914577726216529966, 8.655430955944584362729689025672
