| L(s) = 1 | − 6·3-s + 27·9-s + 96·11-s − 84·17-s − 184·19-s + 146·25-s − 108·27-s − 576·33-s + 12·41-s − 184·43-s − 290·49-s + 504·51-s + 1.10e3·57-s + 1.03e3·59-s + 1.04e3·67-s + 860·73-s − 876·75-s + 405·81-s + 864·83-s − 1.26e3·89-s + 1.72e3·97-s + 2.59e3·99-s + 2.76e3·107-s + 4.21e3·113-s + 4.25e3·121-s − 72·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 2.63·11-s − 1.19·17-s − 2.22·19-s + 1.16·25-s − 0.769·27-s − 3.03·33-s + 0.0457·41-s − 0.652·43-s − 0.845·49-s + 1.38·51-s + 2.56·57-s + 2.27·59-s + 1.91·67-s + 1.37·73-s − 1.34·75-s + 5/9·81-s + 1.14·83-s − 1.50·89-s + 1.80·97-s + 2.63·99-s + 2.49·107-s + 3.50·113-s + 3.19·121-s − 0.0527·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.094971938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.094971938\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 146 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 290 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1942 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 22750 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 48382 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 40178 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 61706 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 206062 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50254 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 516 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 325658 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 524 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 274178 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 430 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 392398 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 432 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 862 T + p^{3} T^{2} )^{2} \) |
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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
−10.08509825993297760030522995205, −9.865084495072012263346090427124, −9.237759324670776086431721628860, −8.856260425524114584414103424322, −8.572367671502278297443722844867, −8.179849312979938799124331207734, −7.25219177704208574854297326674, −6.74192302299485496203504444575, −6.70439005528571524966646096083, −6.33255338170390159385562553662, −5.91527233991602776236110951606, −5.18186339036491029665882696164, −4.55181087589109604276109039114, −4.41826592676586169956660837818, −3.79964579365562355492345370841, −3.38532616526900855820829856775, −2.06240059032154980978798587856, −1.99655074938893176885389896092, −0.966315007837439163114567120413, −0.52789744647342078562359004105,
0.52789744647342078562359004105, 0.966315007837439163114567120413, 1.99655074938893176885389896092, 2.06240059032154980978798587856, 3.38532616526900855820829856775, 3.79964579365562355492345370841, 4.41826592676586169956660837818, 4.55181087589109604276109039114, 5.18186339036491029665882696164, 5.91527233991602776236110951606, 6.33255338170390159385562553662, 6.70439005528571524966646096083, 6.74192302299485496203504444575, 7.25219177704208574854297326674, 8.179849312979938799124331207734, 8.572367671502278297443722844867, 8.856260425524114584414103424322, 9.237759324670776086431721628860, 9.865084495072012263346090427124, 10.08509825993297760030522995205
