| L(s) = 1 | − 4·3-s − 4·5-s − 8·7-s + 6·9-s − 92·11-s − 100·13-s + 16·15-s + 92·17-s − 4·19-s + 32·21-s + 8·23-s − 46·25-s − 92·27-s − 84·29-s − 384·31-s + 368·33-s + 32·35-s + 172·37-s + 400·39-s − 300·41-s − 300·43-s − 24·45-s − 16·47-s − 446·49-s − 368·51-s + 12·53-s + 368·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.357·5-s − 0.431·7-s + 2/9·9-s − 2.52·11-s − 2.13·13-s + 0.275·15-s + 1.31·17-s − 0.0482·19-s + 0.332·21-s + 0.0725·23-s − 0.367·25-s − 0.655·27-s − 0.537·29-s − 2.22·31-s + 1.94·33-s + 0.154·35-s + 0.764·37-s + 1.64·39-s − 1.14·41-s − 1.06·43-s − 0.0795·45-s − 0.0496·47-s − 1.30·49-s − 1.01·51-s + 0.0311·53-s + 0.902·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8 T + 510 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 92 T + 430 p T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 100 T + 5166 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 92 T + 5030 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 11370 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 8798 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 84 T + 45742 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 384 T + 84158 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 172 T + 99294 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 300 T + 157270 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 300 T + 180314 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 114782 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 288382 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 644 T + 462170 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 292 T + 240078 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 172 T + 278250 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 408 T + 672766 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 412 T + 690678 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 400 T + 963870 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 948 T + 1360138 T^{2} + 948 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 572 T + 845846 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2204 T + 2633478 T^{2} - 2204 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.57922595664834922018887791145, −12.23915279818071442399880424868, −11.36404952853558993251899391131, −11.35580316306259526163864712029, −10.26005074571998151693061149626, −10.20565474923032702876951015934, −9.745770621479021276232404444742, −9.010059726596423627200766680499, −7.932237006269247628069458834915, −7.73980837626022015206899727121, −7.34588040132094978657080488008, −6.56637287932644969415958641879, −5.45842832176487131941272740256, −5.37947995676059357712071917456, −4.87922985550077273789094181688, −3.68874375507197308008460118587, −2.88878975924974230801385065637, −2.05498449782590970492279501475, 0, 0,
2.05498449782590970492279501475, 2.88878975924974230801385065637, 3.68874375507197308008460118587, 4.87922985550077273789094181688, 5.37947995676059357712071917456, 5.45842832176487131941272740256, 6.56637287932644969415958641879, 7.34588040132094978657080488008, 7.73980837626022015206899727121, 7.932237006269247628069458834915, 9.010059726596423627200766680499, 9.745770621479021276232404444742, 10.20565474923032702876951015934, 10.26005074571998151693061149626, 11.35580316306259526163864712029, 11.36404952853558993251899391131, 12.23915279818071442399880424868, 12.57922595664834922018887791145
