| L(s) = 1 | − 2·2-s + 3·4-s − 4·7-s − 4·8-s + 9-s + 8·14-s + 5·16-s − 2·18-s + 25-s − 12·28-s − 12·29-s − 6·32-s + 3·36-s + 4·37-s − 8·43-s + 9·49-s − 2·50-s − 12·53-s + 16·56-s + 24·58-s − 4·63-s + 7·64-s − 8·67-s − 4·72-s − 8·74-s + 16·79-s + 81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.51·7-s − 1.41·8-s + 1/3·9-s + 2.13·14-s + 5/4·16-s − 0.471·18-s + 1/5·25-s − 2.26·28-s − 2.22·29-s − 1.06·32-s + 1/2·36-s + 0.657·37-s − 1.21·43-s + 9/7·49-s − 0.282·50-s − 1.64·53-s + 2.13·56-s + 3.15·58-s − 0.503·63-s + 7/8·64-s − 0.977·67-s − 0.471·72-s − 0.929·74-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.626179974771370527778660791757, −9.409651081118525047516189633425, −9.305587122869086271308905422277, −8.379916242317422700081895142791, −7.941231322257043910223245152113, −7.36881236073711860270053595312, −6.83220931658124425661839415899, −6.42217617652666865799421983967, −5.85279772726008439710935686494, −5.13654741900197678322916321630, −3.99226445507565554279020581776, −3.36585804145949552210873743484, −2.61790530319617610939238286369, −1.57147020173234480594288408828, 0,
1.57147020173234480594288408828, 2.61790530319617610939238286369, 3.36585804145949552210873743484, 3.99226445507565554279020581776, 5.13654741900197678322916321630, 5.85279772726008439710935686494, 6.42217617652666865799421983967, 6.83220931658124425661839415899, 7.36881236073711860270053595312, 7.941231322257043910223245152113, 8.379916242317422700081895142791, 9.305587122869086271308905422277, 9.409651081118525047516189633425, 9.626179974771370527778660791757
