| L(s) = 1 | − 8·5-s + 14·7-s − 44·11-s − 60·13-s − 48·17-s + 80·19-s − 140·23-s − 158·25-s + 60·29-s + 232·31-s − 112·35-s − 180·37-s + 344·41-s − 224·43-s − 312·47-s + 147·49-s + 20·53-s + 352·55-s − 64·59-s + 204·61-s + 480·65-s + 872·67-s − 164·71-s + 1.50e3·73-s − 616·77-s + 1.64e3·79-s − 2.20e3·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s + 0.755·7-s − 1.20·11-s − 1.28·13-s − 0.684·17-s + 0.965·19-s − 1.26·23-s − 1.26·25-s + 0.384·29-s + 1.34·31-s − 0.540·35-s − 0.799·37-s + 1.31·41-s − 0.794·43-s − 0.968·47-s + 3/7·49-s + 0.0518·53-s + 0.862·55-s − 0.141·59-s + 0.428·61-s + 0.915·65-s + 1.59·67-s − 0.274·71-s + 2.40·73-s − 0.911·77-s + 2.33·79-s − 2.90·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.141515391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141515391\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T + 222 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 p T + 90 p T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 60 T + 2478 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 48 T + 2966 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 80 T + 15142 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 140 T + 28838 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 60 T + 49502 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 232 T + 72862 T^{2} - 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 180 T + 64350 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 344 T + 107190 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 224 T + 101158 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 312 T + 121982 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 20 T + 69758 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 64 T + 196182 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 204 T + 434622 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 872 T + 790038 T^{2} - 872 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 164 T + 140646 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1500 T + 1319238 T^{2} - 1500 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1640 T + 1417534 T^{2} - 1640 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2200 T + 2283174 T^{2} + 2200 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 264 T + 761686 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2092 T + 2771446 T^{2} - 2092 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.935416709458152496495170063034, −8.341605827869483131153090649011, −8.112959711145092868862503396848, −7.920172716629860278194469054629, −7.49905655433390560686551362582, −7.27404494447259502427751404574, −6.48737210013672922993869208054, −6.46204262417106956096368019427, −5.61359687032693120228160981847, −5.35588564098971071017434716488, −4.92099524505515116160653714102, −4.62290830158455104305915070451, −4.09167163582056596238382403372, −3.71719897013070719973350588490, −3.05470958873537348435560915366, −2.62438020474788316424321783148, −2.01492088614751322152802568408, −1.83163246993822986716459226389, −0.61626426771402527461736222219, −0.47660729317228823848632236287,
0.47660729317228823848632236287, 0.61626426771402527461736222219, 1.83163246993822986716459226389, 2.01492088614751322152802568408, 2.62438020474788316424321783148, 3.05470958873537348435560915366, 3.71719897013070719973350588490, 4.09167163582056596238382403372, 4.62290830158455104305915070451, 4.92099524505515116160653714102, 5.35588564098971071017434716488, 5.61359687032693120228160981847, 6.46204262417106956096368019427, 6.48737210013672922993869208054, 7.27404494447259502427751404574, 7.49905655433390560686551362582, 7.920172716629860278194469054629, 8.112959711145092868862503396848, 8.341605827869483131153090649011, 8.935416709458152496495170063034
