| L(s) = 1 | + 4·2-s + 12·4-s − 14·7-s + 32·8-s − 10·9-s − 56·14-s + 80·16-s − 40·18-s − 20·23-s + 22·25-s − 168·28-s + 192·32-s − 120·36-s − 80·46-s + 147·49-s + 88·50-s − 448·56-s + 140·63-s + 448·64-s − 220·71-s − 320·72-s + 260·79-s + 19·81-s − 240·92-s + 588·98-s + 264·100-s − 1.12e3·112-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s − 2·7-s + 4·8-s − 1.11·9-s − 4·14-s + 5·16-s − 2.22·18-s − 0.869·23-s + 0.879·25-s − 6·28-s + 6·32-s − 3.33·36-s − 1.73·46-s + 3·49-s + 1.75·50-s − 8·56-s + 20/9·63-s + 7·64-s − 3.09·71-s − 4.44·72-s + 3.29·79-s + 0.234·81-s − 2.60·92-s + 6·98-s + 2.63·100-s − 10·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.370444194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.370444194\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 22 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 310 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 650 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1130 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7370 T^{2} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 110 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 13130 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−15.10345366568391284867310541666, −14.73742526128021206439300966693, −13.87100592313024769359403280166, −13.80233634662448084101174921535, −12.94367603906512499983803799601, −12.77803439708070140281812489996, −11.93932197262351958113076719148, −11.79730411118070629160954344729, −10.78120178663063974871041507417, −10.41408750572670792695221940173, −9.641124537596216943263228281486, −8.738979471671570441357073252297, −7.71497155999922153099708490744, −6.95429534079436148206400626163, −6.27128792335923741235043532704, −5.94673847070732305831148657385, −5.09940843239879600097408440845, −3.98297958173207503833937160988, −3.23709374577416374993135913359, −2.57430868159648389432994700692,
2.57430868159648389432994700692, 3.23709374577416374993135913359, 3.98297958173207503833937160988, 5.09940843239879600097408440845, 5.94673847070732305831148657385, 6.27128792335923741235043532704, 6.95429534079436148206400626163, 7.71497155999922153099708490744, 8.738979471671570441357073252297, 9.641124537596216943263228281486, 10.41408750572670792695221940173, 10.78120178663063974871041507417, 11.79730411118070629160954344729, 11.93932197262351958113076719148, 12.77803439708070140281812489996, 12.94367603906512499983803799601, 13.80233634662448084101174921535, 13.87100592313024769359403280166, 14.73742526128021206439300966693, 15.10345366568391284867310541666
