| L(s) = 1 | + 16·4-s + 192·16-s − 112·19-s + 616·31-s + 286·49-s + 364·61-s + 2.04e3·64-s − 1.79e3·76-s − 1.76e3·79-s + 1.29e3·109-s − 2.66e3·121-s + 9.85e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s − 1.35·19-s + 3.56·31-s + 0.833·49-s + 0.764·61-s + 4·64-s − 2.70·76-s − 2.51·79-s + 1.13·109-s − 2·121-s + 7.13·124-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 − 0.230·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.491012790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.491012790\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 89206 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 111386 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 172874 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 638066 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 884 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 56446 T^{2} + p^{6} T^{4} \) |
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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
−11.83317629619339750275411306096, −11.54220132617010680369072812269, −11.27106770442625783710335372703, −10.48643982641229224489160539567, −10.18413032330575647567067529043, −10.05937957131318552609303742528, −9.008535564018519969191632595058, −8.418711145578016408013338854172, −8.001600599237077787702217421772, −7.52623313079840970827005365401, −6.69011148840149696759225328526, −6.64584941265074391384830175252, −6.05506834780535245079356182681, −5.49000046502699681644352355077, −4.57589828502556953704423015645, −3.91851140829057833404866719228, −2.87111924768286360602338227764, −2.62208392113583812603295536198, −1.77300373682735636680133638761, −0.887581425520046305461421266470,
0.887581425520046305461421266470, 1.77300373682735636680133638761, 2.62208392113583812603295536198, 2.87111924768286360602338227764, 3.91851140829057833404866719228, 4.57589828502556953704423015645, 5.49000046502699681644352355077, 6.05506834780535245079356182681, 6.64584941265074391384830175252, 6.69011148840149696759225328526, 7.52623313079840970827005365401, 8.001600599237077787702217421772, 8.418711145578016408013338854172, 9.008535564018519969191632595058, 10.05937957131318552609303742528, 10.18413032330575647567067529043, 10.48643982641229224489160539567, 11.27106770442625783710335372703, 11.54220132617010680369072812269, 11.83317629619339750275411306096
