| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s + 6·5-s + 6·6-s − 3·8-s + 3·9-s − 18·10-s − 2·11-s − 8·12-s − 12·15-s + 3·16-s + 3·17-s − 9·18-s + 5·19-s + 24·20-s + 6·22-s − 5·23-s + 6·24-s + 17·25-s − 4·27-s − 11·29-s + 36·30-s + 7·31-s − 6·32-s + 4·33-s − 9·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s + 2.68·5-s + 2.44·6-s − 1.06·8-s + 9-s − 5.69·10-s − 0.603·11-s − 2.30·12-s − 3.09·15-s + 3/4·16-s + 0.727·17-s − 2.12·18-s + 1.14·19-s + 5.36·20-s + 1.27·22-s − 1.04·23-s + 1.22·24-s + 17/5·25-s − 0.769·27-s − 2.04·29-s + 6.57·30-s + 1.25·31-s − 1.06·32-s + 0.696·33-s − 1.54·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75220929 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75220929 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7379066862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7379066862\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 59 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 51 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 11 T + 3 p T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 73 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 51 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 85 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 133 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 133 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 141 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 121 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 175 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 147 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 199 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.71497403740230538730075730255, −7.71135668676448683131166946704, −7.30978946309367927068872545662, −7.23787646954009108749072960160, −6.26224114105384917204530355217, −6.19335078394691205142117571541, −6.07621856411657874394924906588, −5.57897865692508412682323859772, −5.57668980936888796884375075709, −4.96878687225473718565715971398, −4.73675929897362230986449072289, −4.13174992398160337816013969668, −3.43912190748422268442908556647, −3.15282127242341154530069099977, −2.38309318546117603536466016310, −2.14378250282753644018431845573, −1.80935935184040116142170138062, −1.19859717345043575729236220870, −1.02989656765780724568060027345, −0.36381876509405884661402964895,
0.36381876509405884661402964895, 1.02989656765780724568060027345, 1.19859717345043575729236220870, 1.80935935184040116142170138062, 2.14378250282753644018431845573, 2.38309318546117603536466016310, 3.15282127242341154530069099977, 3.43912190748422268442908556647, 4.13174992398160337816013969668, 4.73675929897362230986449072289, 4.96878687225473718565715971398, 5.57668980936888796884375075709, 5.57897865692508412682323859772, 6.07621856411657874394924906588, 6.19335078394691205142117571541, 6.26224114105384917204530355217, 7.23787646954009108749072960160, 7.30978946309367927068872545662, 7.71135668676448683131166946704, 7.71497403740230538730075730255
