| L(s) = 1 | + 2.79e4·3-s + 1.22e6·5-s − 8.85e7·7-s − 4.01e8·9-s + 7.16e9·11-s − 1.01e10·13-s + 3.42e10·15-s − 7.20e10·17-s + 3.12e12·19-s − 2.47e12·21-s + 1.47e13·23-s − 3.44e13·25-s − 1.16e13·27-s − 3.02e13·29-s + 1.23e14·31-s + 1.99e14·33-s − 1.08e14·35-s + 2.01e15·37-s − 2.82e14·39-s + 2.54e15·41-s + 5.63e15·43-s − 4.92e14·45-s + 2.19e16·47-s − 3.29e15·49-s − 2.01e15·51-s − 9.41e15·53-s + 8.78e15·55-s + ⋯ |
| L(s) = 1 | + 0.818·3-s + 0.280·5-s − 0.829·7-s − 0.345·9-s + 0.916·11-s − 0.264·13-s + 0.229·15-s − 0.147·17-s + 2.21·19-s − 0.678·21-s + 1.70·23-s − 1.80·25-s − 0.295·27-s − 0.387·29-s + 0.838·31-s + 0.749·33-s − 0.232·35-s + 2.54·37-s − 0.216·39-s + 1.21·41-s + 1.70·43-s − 0.0969·45-s + 2.86·47-s − 0.289·49-s − 0.120·51-s − 0.392·53-s + 0.257·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(4.573605616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.573605616\) |
| \(L(\frac{21}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 9304 p T + 14570470 p^{4} T^{2} - 9304 p^{20} T^{3} + p^{38} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 245324 p T + 1437226145582 p^{2} T^{2} - 245324 p^{20} T^{3} + p^{38} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 88510512 T + 1589951031584546 p T^{2} + 88510512 p^{19} T^{3} + p^{38} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7163787608 T + 6207054254616739618 p T^{2} - 7163787608 p^{19} T^{3} + p^{38} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10126923604 T + \)\(22\!\cdots\!66\)\( p T^{2} + 10126923604 p^{19} T^{3} + p^{38} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4237945820 p T + \)\(26\!\cdots\!54\)\( p^{2} T^{2} + 4237945820 p^{20} T^{3} + p^{38} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 164235814328 p T + \)\(54\!\cdots\!14\)\( T^{2} - 164235814328 p^{20} T^{3} + p^{38} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 14759207090288 T + \)\(68\!\cdots\!70\)\( p T^{2} - 14759207090288 p^{19} T^{3} + p^{38} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 30249539245044 T + \)\(11\!\cdots\!22\)\( T^{2} + 30249539245044 p^{19} T^{3} + p^{38} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 123389562777920 T + \)\(27\!\cdots\!42\)\( T^{2} - 123389562777920 p^{19} T^{3} + p^{38} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2015393170174524 T + \)\(60\!\cdots\!70\)\( p T^{2} - 2015393170174524 p^{19} T^{3} + p^{38} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2540784959504244 T + \)\(74\!\cdots\!06\)\( T^{2} - 2540784959504244 p^{19} T^{3} + p^{38} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5633655093389464 T + \)\(29\!\cdots\!38\)\( T^{2} - 5633655093389464 p^{19} T^{3} + p^{38} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21948339587130336 T + \)\(23\!\cdots\!90\)\( T^{2} - 21948339587130336 p^{19} T^{3} + p^{38} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9418125066904676 T + \)\(29\!\cdots\!78\)\( T^{2} + 9418125066904676 p^{19} T^{3} + p^{38} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 98542449590407624 T + \)\(95\!\cdots\!22\)\( T^{2} + 98542449590407624 p^{19} T^{3} + p^{38} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10292145377839820 T + \)\(11\!\cdots\!82\)\( T^{2} - 10292145377839820 p^{19} T^{3} + p^{38} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 75753628003984504 T + \)\(44\!\cdots\!10\)\( T^{2} + 75753628003984504 p^{19} T^{3} + p^{38} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 17407052566713776 T + \)\(27\!\cdots\!06\)\( T^{2} + 17407052566713776 p^{19} T^{3} + p^{38} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 857508255059832268 T + \)\(53\!\cdots\!30\)\( T^{2} + 857508255059832268 p^{19} T^{3} + p^{38} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 226291921444855072 T + \)\(21\!\cdots\!34\)\( T^{2} - 226291921444855072 p^{19} T^{3} + p^{38} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 767515701460985048 T + \)\(59\!\cdots\!70\)\( T^{2} + 767515701460985048 p^{19} T^{3} + p^{38} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6092545894435174548 T + \)\(31\!\cdots\!94\)\( T^{2} - 6092545894435174548 p^{19} T^{3} + p^{38} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1548148249522347076 T + \)\(66\!\cdots\!10\)\( T^{2} - 1548148249522347076 p^{19} T^{3} + p^{38} T^{4} \) |
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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
−14.62388010567234321111733381743, −14.27037415368070673721261709048, −13.58552085288741947062518985988, −13.12131423987567914270192426301, −12.15714655604874204099006197117, −11.59914081415668636888883286447, −10.81928787004886232591321745195, −9.639204849291178637420501281352, −9.394296198704092137514479202395, −8.928425317591756538544544587016, −7.60547564123989661027014542665, −7.43732047132355347772932816333, −6.12667891270838293501618924351, −5.77607192252857313240520596880, −4.53779150121156183751982221043, −3.70758560218737834722701370721, −2.85702721071352002210998223132, −2.53869078805911618248018401187, −1.20176097278202864849037478207, −0.67878721940818822302300416822,
0.67878721940818822302300416822, 1.20176097278202864849037478207, 2.53869078805911618248018401187, 2.85702721071352002210998223132, 3.70758560218737834722701370721, 4.53779150121156183751982221043, 5.77607192252857313240520596880, 6.12667891270838293501618924351, 7.43732047132355347772932816333, 7.60547564123989661027014542665, 8.928425317591756538544544587016, 9.394296198704092137514479202395, 9.639204849291178637420501281352, 10.81928787004886232591321745195, 11.59914081415668636888883286447, 12.15714655604874204099006197117, 13.12131423987567914270192426301, 13.58552085288741947062518985988, 14.27037415368070673721261709048, 14.62388010567234321111733381743
