| L(s) = 1 | − 2-s + 2·3-s − 2·4-s − 2·6-s − 6·7-s + 3·8-s + 3·9-s − 4·12-s − 3·13-s + 6·14-s + 16-s − 6·17-s − 3·18-s + 10·19-s − 12·21-s + 13·23-s + 6·24-s + 3·26-s + 4·27-s + 12·28-s − 5·29-s − 11·31-s − 2·32-s + 6·34-s − 6·36-s + 11·37-s − 10·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s − 2.26·7-s + 1.06·8-s + 9-s − 1.15·12-s − 0.832·13-s + 1.60·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 2.29·19-s − 2.61·21-s + 2.71·23-s + 1.22·24-s + 0.588·26-s + 0.769·27-s + 2.26·28-s − 0.928·29-s − 1.97·31-s − 0.353·32-s + 1.02·34-s − 36-s + 1.80·37-s − 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 13 T + 87 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 81 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 21 T + 203 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 163 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 181 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 147 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 23 T + 297 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.39401641156667914957717249581, −7.24107463384482650229178432176, −7.00825528403734035525457473978, −6.93099265010900980073454544313, −6.33178366059962705640267396890, −5.75895666993312151963607594144, −5.36913166120561568995693164280, −5.32203609987031913940876369658, −4.57263624081194192948159330194, −4.43148619366494107008439830845, −3.75688340154129936556983250799, −3.73465289954250653423028340984, −3.10429365087865373262365620223, −2.89388583190641957976203503624, −2.62443962389091585188920222834, −2.08335802483974121077697791137, −1.14542483479338963190160464703, −1.04902840596070642649305272918, 0, 0,
1.04902840596070642649305272918, 1.14542483479338963190160464703, 2.08335802483974121077697791137, 2.62443962389091585188920222834, 2.89388583190641957976203503624, 3.10429365087865373262365620223, 3.73465289954250653423028340984, 3.75688340154129936556983250799, 4.43148619366494107008439830845, 4.57263624081194192948159330194, 5.32203609987031913940876369658, 5.36913166120561568995693164280, 5.75895666993312151963607594144, 6.33178366059962705640267396890, 6.93099265010900980073454544313, 7.00825528403734035525457473978, 7.24107463384482650229178432176, 7.39401641156667914957717249581
