| L(s) = 1 | − 4·5-s − 2·7-s + 4·11-s + 2·13-s + 2·17-s − 2·19-s − 6·23-s + 2·25-s + 10·29-s + 6·31-s + 8·35-s − 10·37-s + 8·43-s + 3·49-s + 2·53-s − 16·55-s − 4·59-s − 4·61-s − 8·65-s − 12·67-s + 6·71-s − 20·73-s − 8·77-s + 18·79-s − 10·83-s − 8·85-s + 16·89-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.458·19-s − 1.25·23-s + 2/5·25-s + 1.85·29-s + 1.07·31-s + 1.35·35-s − 1.64·37-s + 1.21·43-s + 3/7·49-s + 0.274·53-s − 2.15·55-s − 0.520·59-s − 0.512·61-s − 0.992·65-s − 1.46·67-s + 0.712·71-s − 2.34·73-s − 0.911·77-s + 2.02·79-s − 1.09·83-s − 0.867·85-s + 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.595926365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595926365\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−7.84951657561159274106672978736, −7.60358830764236590411092016299, −7.07474396074337534150988546293, −6.99337560207161330067812098486, −6.39649675499337916231273539640, −6.31480450206719184619501330049, −5.79645148908963802216614021230, −5.75058348493841252944320472383, −4.84588209817386277044704354395, −4.66293939515861899167726718694, −4.26791466512881524167539492356, −3.97319549997191632117540244165, −3.58789587897799903431276403506, −3.51297941035596890293402275489, −2.82970041661384375884560329137, −2.66144195653829354079566755847, −1.76980991153540768376863713602, −1.52196817292305346432031620228, −0.70153349473876257868841530637, −0.42607731870913130939555465244,
0.42607731870913130939555465244, 0.70153349473876257868841530637, 1.52196817292305346432031620228, 1.76980991153540768376863713602, 2.66144195653829354079566755847, 2.82970041661384375884560329137, 3.51297941035596890293402275489, 3.58789587897799903431276403506, 3.97319549997191632117540244165, 4.26791466512881524167539492356, 4.66293939515861899167726718694, 4.84588209817386277044704354395, 5.75058348493841252944320472383, 5.79645148908963802216614021230, 6.31480450206719184619501330049, 6.39649675499337916231273539640, 6.99337560207161330067812098486, 7.07474396074337534150988546293, 7.60358830764236590411092016299, 7.84951657561159274106672978736
