| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s − 8·11-s + 2·12-s + 16-s − 4·17-s + 3·18-s + 8·19-s − 8·22-s + 2·24-s − 6·25-s + 4·27-s + 32-s − 16·33-s − 4·34-s + 3·36-s + 8·38-s + 12·41-s − 8·43-s − 8·44-s + 2·48-s − 6·50-s − 8·51-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 9-s − 2.41·11-s + 0.577·12-s + 1/4·16-s − 0.970·17-s + 0.707·18-s + 1.83·19-s − 1.70·22-s + 0.408·24-s − 6/5·25-s + 0.769·27-s + 0.176·32-s − 2.78·33-s − 0.685·34-s + 1/2·36-s + 1.29·38-s + 1.87·41-s − 1.21·43-s − 1.20·44-s + 0.288·48-s − 0.848·50-s − 1.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2765952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2765952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.429102453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.429102453\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−7.68532191797102726979476428936, −7.28965766970175880058840390576, −6.95470194591461271128583103192, −6.23175022246340560839371166306, −5.81340647648643732482215027480, −5.47990216984766844713395370376, −4.82857955948335988380640999352, −4.73982413848000625426104276677, −4.09458451974208041146506029777, −3.47205407811273829747507807541, −3.08481257245522303063597583848, −2.74481126065377142074711550653, −2.16063348217202322480816158281, −1.79260226536651551834145289140, −0.62964118914525720642781422582,
0.62964118914525720642781422582, 1.79260226536651551834145289140, 2.16063348217202322480816158281, 2.74481126065377142074711550653, 3.08481257245522303063597583848, 3.47205407811273829747507807541, 4.09458451974208041146506029777, 4.73982413848000625426104276677, 4.82857955948335988380640999352, 5.47990216984766844713395370376, 5.81340647648643732482215027480, 6.23175022246340560839371166306, 6.95470194591461271128583103192, 7.28965766970175880058840390576, 7.68532191797102726979476428936
