L(s) = 1 | − 24·5-s − 100·13-s − 24·17-s + 288·25-s − 624·29-s + 316·37-s + 1.27e3·49-s − 1.39e3·53-s − 932·61-s + 2.40e3·65-s − 81·81-s + 576·85-s + 6.38e3·97-s − 2.20e3·101-s − 1.77e3·109-s + 4.24e3·113-s − 4.72e3·125-s + 127-s + 131-s + 137-s + 139-s + 1.49e4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 2.14·5-s − 2.13·13-s − 0.342·17-s + 2.30·25-s − 3.99·29-s + 1.40·37-s + 3.70·49-s − 3.60·53-s − 1.95·61-s + 4.57·65-s − 1/9·81-s + 0.735·85-s + 6.68·97-s − 2.17·101-s − 1.55·109-s + 3.53·113-s − 3.38·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 8.57·145-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.04791669040\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04791669040\) |
\(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
good | 5 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 636 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 2797202 T^{4} + p^{12} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 50 T + 1250 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 + 93105362 T^{4} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 23686 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 312 T + 48672 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 33132 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 158 T + 12482 T^{2} - 158 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 137806 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 11507170894 T^{4} + p^{12} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 205846 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 696 T + 242208 T^{2} + 696 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 15033124078 T^{4} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 466 T + 108578 T^{2} + 466 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 168637908238 T^{4} + p^{12} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 10150 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 591410 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 888396 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 425666259214 T^{4} + p^{12} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 151054 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1596 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.13346080489982483850158446034, −6.98473674655187567281676516447, −6.66539835472985655810466450817, −6.20001358126985791884612664625, −6.04637976390616640777278890056, −5.78796254105345388669722391971, −5.78142162632866773519807669540, −5.11959114195127008067987364481, −5.09264365578713533991868158948, −4.65502387408547479348982377521, −4.64956606729124854289052076911, −4.44860478026395446496726810363, −3.99807357516786941563963282046, −3.78905103973694593793876958282, −3.59727511283480016112244930079, −3.36815798576158829887023343976, −3.10044942894055618368602948782, −2.55664386256607323547158926299, −2.45285550064711594048839060382, −2.05656524730146270355832266474, −1.86817654563085275249904550742, −1.24052347261788815162274629451, −0.910820011162474349493899022140, −0.32330259118935361848728003881, −0.06396516021725235218822030762,
0.06396516021725235218822030762, 0.32330259118935361848728003881, 0.910820011162474349493899022140, 1.24052347261788815162274629451, 1.86817654563085275249904550742, 2.05656524730146270355832266474, 2.45285550064711594048839060382, 2.55664386256607323547158926299, 3.10044942894055618368602948782, 3.36815798576158829887023343976, 3.59727511283480016112244930079, 3.78905103973694593793876958282, 3.99807357516786941563963282046, 4.44860478026395446496726810363, 4.64956606729124854289052076911, 4.65502387408547479348982377521, 5.09264365578713533991868158948, 5.11959114195127008067987364481, 5.78142162632866773519807669540, 5.78796254105345388669722391971, 6.04637976390616640777278890056, 6.20001358126985791884612664625, 6.66539835472985655810466450817, 6.98473674655187567281676516447, 7.13346080489982483850158446034