| L(s) = 1 | + 2·3-s − 3·5-s + 2·7-s + 3·9-s + 2·11-s + 2·13-s − 6·15-s + 6·17-s − 19-s + 4·21-s − 5·23-s + 25-s + 4·27-s + 5·29-s + 7·31-s + 4·33-s − 6·35-s − 6·37-s + 4·39-s − 4·41-s + 15·43-s − 9·45-s − 15·47-s + 3·49-s + 12·51-s + 53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s + 0.755·7-s + 9-s + 0.603·11-s + 0.554·13-s − 1.54·15-s + 1.45·17-s − 0.229·19-s + 0.872·21-s − 1.04·23-s + 1/5·25-s + 0.769·27-s + 0.928·29-s + 1.25·31-s + 0.696·33-s − 1.01·35-s − 0.986·37-s + 0.640·39-s − 0.624·41-s + 2.28·43-s − 1.34·45-s − 2.18·47-s + 3/7·49-s + 1.68·51-s + 0.137·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.603031334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.603031334\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_4$ | \( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 68 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 21 T + 252 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 126 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 25 T + 318 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−8.428732870120143246997077716568, −8.162961683798306201528927875652, −7.86179825681772285450318564842, −7.72785165260807516924028859495, −7.21471302344512570429017372836, −6.93002299136013074874690558475, −6.27824612123655113956990913055, −6.24439650830023806252952381379, −5.44620018368400155353800918340, −5.24369371444085294707130686793, −4.49898879312328056471756123810, −4.39215049353693915661974230681, −3.86555700017526091287815445132, −3.72625779458556131655850660409, −3.03454878368865648297119705093, −3.01730581313045701474129993398, −2.09163302897418702891970208106, −1.80627805218947564472894508280, −1.10906969638255060049565103562, −0.63284475997362420582984389504,
0.63284475997362420582984389504, 1.10906969638255060049565103562, 1.80627805218947564472894508280, 2.09163302897418702891970208106, 3.01730581313045701474129993398, 3.03454878368865648297119705093, 3.72625779458556131655850660409, 3.86555700017526091287815445132, 4.39215049353693915661974230681, 4.49898879312328056471756123810, 5.24369371444085294707130686793, 5.44620018368400155353800918340, 6.24439650830023806252952381379, 6.27824612123655113956990913055, 6.93002299136013074874690558475, 7.21471302344512570429017372836, 7.72785165260807516924028859495, 7.86179825681772285450318564842, 8.162961683798306201528927875652, 8.428732870120143246997077716568
