L(s) = 1 | − 20·13-s − 48·25-s − 48·37-s − 98·49-s + 240·61-s − 192·73-s + 288·97-s − 364·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | − 1.53·13-s − 1.91·25-s − 1.29·37-s − 2·49-s + 3.93·61-s − 2.63·73-s + 2.96·97-s − 3.33·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.224·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2675568326\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2675568326\) |
\(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 48 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 480 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 1680 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1440 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5040 T^{2} + p^{4} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12480 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{2} \) |
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
−9.695855784929875814110270719981, −8.564634579849044980618849756050, −8.258583889709116907322130903071, −7.81168594740767994173515429461, −7.55735621592454448647834806651, −6.97767996621339476319185480452, −6.87972779688511387690117164584, −6.35693005428886214399257719511, −5.67994516340178108389434591084, −5.59764050585310530830430213612, −5.06894533191213431218411588291, −4.58851345151209340250416713555, −4.30290249658858864135945210930, −3.54857571936561452205448044510, −3.43665805238807338015252311561, −2.66085326806900251101363509893, −2.15601233509950635000065410369, −1.86975634581687124417112877302, −1.06217249926952351275689743864, −0.13208250704944538916571866129,
0.13208250704944538916571866129, 1.06217249926952351275689743864, 1.86975634581687124417112877302, 2.15601233509950635000065410369, 2.66085326806900251101363509893, 3.43665805238807338015252311561, 3.54857571936561452205448044510, 4.30290249658858864135945210930, 4.58851345151209340250416713555, 5.06894533191213431218411588291, 5.59764050585310530830430213612, 5.67994516340178108389434591084, 6.35693005428886214399257719511, 6.87972779688511387690117164584, 6.97767996621339476319185480452, 7.55735621592454448647834806651, 7.81168594740767994173515429461, 8.258583889709116907322130903071, 8.564634579849044980618849756050, 9.695855784929875814110270719981