| L(s) = 1 | + 2·5-s − 2·11-s − 4·13-s + 6·17-s − 4·19-s − 6·23-s + 5·25-s − 4·31-s − 10·37-s − 4·41-s − 8·43-s + 4·47-s + 12·53-s − 4·55-s + 12·59-s + 6·61-s − 8·65-s + 4·67-s − 28·71-s − 2·73-s + 8·79-s + 32·83-s + 12·85-s − 6·89-s − 8·95-s + 36·97-s + 14·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.603·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 25-s − 0.718·31-s − 1.64·37-s − 0.624·41-s − 1.21·43-s + 0.583·47-s + 1.64·53-s − 0.539·55-s + 1.56·59-s + 0.768·61-s − 0.992·65-s + 0.488·67-s − 3.32·71-s − 0.234·73-s + 0.900·79-s + 3.51·83-s + 1.30·85-s − 0.635·89-s − 0.820·95-s + 3.65·97-s + 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.750939582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750939582\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.681527256123848456916835176163, −8.456915672812888172534004603436, −8.061994463765503089967364434994, −7.45652547754273956512006153005, −7.40466563012202686079955700986, −6.99294442133662316984318548358, −6.32461466273429206218829640833, −6.28760488521362731229621087702, −5.66938616783286731043020482749, −5.29026900721562383130903797779, −5.10688092101081822765516915254, −4.75932387608012061385097593670, −4.04876327244016960146224424850, −3.62441852098118416003424704394, −3.31285065771985867458379327118, −2.63980110593827424901677498510, −2.09736509449675253963484815553, −2.03615360149645158976430055265, −1.22384112172813856586386458748, −0.39515998465855833571210663598,
0.39515998465855833571210663598, 1.22384112172813856586386458748, 2.03615360149645158976430055265, 2.09736509449675253963484815553, 2.63980110593827424901677498510, 3.31285065771985867458379327118, 3.62441852098118416003424704394, 4.04876327244016960146224424850, 4.75932387608012061385097593670, 5.10688092101081822765516915254, 5.29026900721562383130903797779, 5.66938616783286731043020482749, 6.28760488521362731229621087702, 6.32461466273429206218829640833, 6.99294442133662316984318548358, 7.40466563012202686079955700986, 7.45652547754273956512006153005, 8.061994463765503089967364434994, 8.456915672812888172534004603436, 8.681527256123848456916835176163
