| L(s) = 1 | + 4·2-s + 8·4-s − 5·7-s + 8·8-s − 2·11-s − 20·14-s − 4·16-s − 8·22-s + 8·23-s − 25-s − 40·28-s + 4·29-s − 32·32-s − 2·43-s − 16·44-s + 32·46-s + 18·49-s − 4·50-s − 20·53-s − 40·56-s + 16·58-s − 64·64-s + 16·67-s + 24·71-s + 10·77-s + 32·79-s − 8·86-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s − 1.88·7-s + 2.82·8-s − 0.603·11-s − 5.34·14-s − 16-s − 1.70·22-s + 1.66·23-s − 1/5·25-s − 7.55·28-s + 0.742·29-s − 5.65·32-s − 0.304·43-s − 2.41·44-s + 4.71·46-s + 18/7·49-s − 0.565·50-s − 2.74·53-s − 5.34·56-s + 2.10·58-s − 8·64-s + 1.95·67-s + 2.84·71-s + 1.13·77-s + 3.60·79-s − 0.862·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.370768440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.370768440\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.85582869646283906731190811336, −6.85073886350288365363080648917, −6.77124936049535641641720020484, −6.54381789640621822379739731453, −6.12241751609850681792173625996, −5.42089478131328865490166156321, −5.34668146252658900228619106104, −4.80004704636396746296312994207, −4.43252527426558787350954315583, −3.67840796781373175574076394185, −3.48189187269012783645528672334, −3.11587306879081144724928834922, −2.61687998381550726785939862479, −2.15535598087150874841625030639, −0.58876266921858160851854225947,
0.58876266921858160851854225947, 2.15535598087150874841625030639, 2.61687998381550726785939862479, 3.11587306879081144724928834922, 3.48189187269012783645528672334, 3.67840796781373175574076394185, 4.43252527426558787350954315583, 4.80004704636396746296312994207, 5.34668146252658900228619106104, 5.42089478131328865490166156321, 6.12241751609850681792173625996, 6.54381789640621822379739731453, 6.77124936049535641641720020484, 6.85073886350288365363080648917, 7.85582869646283906731190811336
