L(s) = 1 | − 90·9-s − 264·11-s + 1.00e3·19-s − 4.38e3·29-s − 4.62e3·31-s + 2.48e3·41-s + 4.03e3·49-s + 1.59e4·59-s + 3.32e4·61-s + 4.90e4·71-s − 9.24e4·79-s − 5.09e4·81-s + 2.20e5·89-s + 2.37e4·99-s + 2.83e5·101-s + 3.49e5·109-s − 2.69e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.52e5·169-s + ⋯ |
L(s) = 1 | − 0.370·9-s − 0.657·11-s + 0.635·19-s − 0.967·29-s − 0.864·31-s + 0.230·41-s + 0.239·49-s + 0.596·59-s + 1.14·61-s + 1.15·71-s − 1.66·79-s − 0.862·81-s + 2.95·89-s + 0.243·99-s + 2.76·101-s + 2.81·109-s − 1.67·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 0.410·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.769960301\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.769960301\) |
\(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 + 10 p^{2} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4030 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 p T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 152330 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2790430 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 500 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 170590 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2190 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2312 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 12305350 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1242 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 131332490 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 415288270 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 392614630 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7980 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 16622 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2696981350 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 24528 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3726958510 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 46240 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 5217997510 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 110310 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11030942590 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.20126901418439867195428815207, −10.11774491610350593784733003098, −9.935665805690466314738188326736, −9.260495622751062300999562447070, −8.838173691090341364202998843199, −8.504310067211096401092354883041, −7.79526988517850202405748539369, −7.43964408757535550862562883538, −7.14264599034062332950794760910, −6.28528227361147548101432668806, −5.97868768977696243609796420838, −5.22499422355608568090279560161, −5.12393732157512081686439900406, −4.27308816015194153223599460256, −3.61203156203784594417808070061, −3.22064386513993205444521383317, −2.38211939212763706650181030381, −1.97227214758327997329873411582, −1.03103811712080657343912961905, −0.36712056391158025581312131368,
0.36712056391158025581312131368, 1.03103811712080657343912961905, 1.97227214758327997329873411582, 2.38211939212763706650181030381, 3.22064386513993205444521383317, 3.61203156203784594417808070061, 4.27308816015194153223599460256, 5.12393732157512081686439900406, 5.22499422355608568090279560161, 5.97868768977696243609796420838, 6.28528227361147548101432668806, 7.14264599034062332950794760910, 7.43964408757535550862562883538, 7.79526988517850202405748539369, 8.504310067211096401092354883041, 8.838173691090341364202998843199, 9.260495622751062300999562447070, 9.935665805690466314738188326736, 10.11774491610350593784733003098, 11.20126901418439867195428815207