| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 12-s + 12·13-s − 16-s + 18-s + 8·23-s − 3·24-s − 6·25-s + 12·26-s + 27-s + 5·32-s − 36-s − 20·37-s + 12·39-s + 8·46-s + 24·47-s − 48-s − 14·49-s − 6·50-s − 12·52-s + 54-s − 24·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.288·12-s + 3.32·13-s − 1/4·16-s + 0.235·18-s + 1.66·23-s − 0.612·24-s − 6/5·25-s + 2.35·26-s + 0.192·27-s + 0.883·32-s − 1/6·36-s − 3.28·37-s + 1.92·39-s + 1.17·46-s + 3.50·47-s − 0.144·48-s − 2·49-s − 0.848·50-s − 1.66·52-s + 0.136·54-s − 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.720199700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.720199700\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.092693734190788229037074202448, −9.032116632714081125535639739797, −8.263513432862467471567669978548, −8.062008804214737453511932083588, −7.33667434445098454643296446245, −6.46787589567348650226556085189, −6.39584814801769229586264060019, −5.69178884171323828176864961556, −5.22523151161492305197654101068, −4.56991321080597356703354311172, −3.84859941092912106278451327058, −3.44365518362789223021999993571, −3.25148783575260438600086430315, −1.95255670636420425042320954209, −1.07628396925620245570845051253,
1.07628396925620245570845051253, 1.95255670636420425042320954209, 3.25148783575260438600086430315, 3.44365518362789223021999993571, 3.84859941092912106278451327058, 4.56991321080597356703354311172, 5.22523151161492305197654101068, 5.69178884171323828176864961556, 6.39584814801769229586264060019, 6.46787589567348650226556085189, 7.33667434445098454643296446245, 8.062008804214737453511932083588, 8.263513432862467471567669978548, 9.032116632714081125535639739797, 9.092693734190788229037074202448
