| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s + 2·7-s + 4·8-s + 8·10-s − 2·13-s + 4·14-s + 8·16-s − 4·17-s − 6·19-s + 8·20-s − 10·23-s + 5·25-s − 4·26-s + 4·28-s + 6·29-s + 8·31-s + 8·32-s − 8·34-s + 8·35-s − 12·37-s − 12·38-s + 16·40-s + 8·41-s + 6·43-s − 20·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s + 0.755·7-s + 1.41·8-s + 2.52·10-s − 0.554·13-s + 1.06·14-s + 2·16-s − 0.970·17-s − 1.37·19-s + 1.78·20-s − 2.08·23-s + 25-s − 0.784·26-s + 0.755·28-s + 1.11·29-s + 1.43·31-s + 1.41·32-s − 1.37·34-s + 1.35·35-s − 1.97·37-s − 1.94·38-s + 2.52·40-s + 1.24·41-s + 0.914·43-s − 2.94·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.60343467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.60343467\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 83 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 104 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 4 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 168 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 199 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 139 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 216 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.82271697659443327769773234159, −7.63736919324340261613472559676, −7.49246120516838914451605549742, −6.55895320892468116108277198285, −6.50602127941327054817591244639, −6.34576238036369101376869259730, −5.80486548497530681074414475484, −5.72699612746372057769697907446, −5.00712215030050945117590091581, −4.95713381034139228636003308805, −4.47735199066914661561583178891, −4.41796077367318607630670100781, −3.71158211495319005647780923355, −3.66260622740656559080852838232, −2.62279026696080044199097281026, −2.50715559904232649040082733645, −2.15534812477300282918118396614, −1.67925267629047420391650953521, −1.47879916872234585177897894189, −0.53611777412567401134703813685,
0.53611777412567401134703813685, 1.47879916872234585177897894189, 1.67925267629047420391650953521, 2.15534812477300282918118396614, 2.50715559904232649040082733645, 2.62279026696080044199097281026, 3.66260622740656559080852838232, 3.71158211495319005647780923355, 4.41796077367318607630670100781, 4.47735199066914661561583178891, 4.95713381034139228636003308805, 5.00712215030050945117590091581, 5.72699612746372057769697907446, 5.80486548497530681074414475484, 6.34576238036369101376869259730, 6.50602127941327054817591244639, 6.55895320892468116108277198285, 7.49246120516838914451605549742, 7.63736919324340261613472559676, 7.82271697659443327769773234159
