| L(s) = 1 | + 10·5-s − 56·11-s − 120·19-s − 25·25-s + 180·29-s + 256·31-s − 484·41-s + 10·49-s − 560·55-s + 40·59-s + 1.08e3·61-s − 2.25e3·71-s − 1.44e3·79-s − 980·89-s − 1.20e3·95-s + 1.15e3·101-s − 740·109-s − 310·121-s − 1.50e3·125-s + 127-s + 131-s + 137-s + 139-s + 1.80e3·145-s + 149-s + 151-s + 2.56e3·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.53·11-s − 1.44·19-s − 1/5·25-s + 1.15·29-s + 1.48·31-s − 1.84·41-s + 0.0291·49-s − 1.37·55-s + 0.0882·59-s + 2.27·61-s − 3.77·71-s − 2.05·79-s − 1.16·89-s − 1.29·95-s + 1.13·101-s − 0.650·109-s − 0.232·121-s − 1.07·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1.03·145-s + 0.000549·149-s + 0.000538·151-s + 1.32·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.230156505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230156505\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5730 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 60 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 128 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 45610 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 242 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 27970 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 156570 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 286090 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 542 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 413170 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1128 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 378610 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 720 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 915090 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 490 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 294590 T^{2} + p^{6} T^{4} \) |
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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
−10.34581645352371701977329874842, −9.816286128502080798848836558359, −9.646052268135818892784560341564, −8.695598428592748740083714589366, −8.437400582584077383016130373626, −8.380848014452515487858073964046, −7.64506717155168582949716136865, −7.06836026275220779523099613789, −6.77120246454101472728929849329, −6.04820468977411626267255619053, −5.92820828614388235569216177166, −5.29149880757721249256393405776, −4.77629299257984150591043866522, −4.43761069914329256552356823254, −3.72567478845097658737359769558, −2.77194529621760488438117723485, −2.71373281496499454479438303591, −1.95500920297652367921591402519, −1.32112351694799658375442404034, −0.30037764341700008923397956321,
0.30037764341700008923397956321, 1.32112351694799658375442404034, 1.95500920297652367921591402519, 2.71373281496499454479438303591, 2.77194529621760488438117723485, 3.72567478845097658737359769558, 4.43761069914329256552356823254, 4.77629299257984150591043866522, 5.29149880757721249256393405776, 5.92820828614388235569216177166, 6.04820468977411626267255619053, 6.77120246454101472728929849329, 7.06836026275220779523099613789, 7.64506717155168582949716136865, 8.380848014452515487858073964046, 8.437400582584077383016130373626, 8.695598428592748740083714589366, 9.646052268135818892784560341564, 9.816286128502080798848836558359, 10.34581645352371701977329874842
