| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 4·6-s − 3·9-s − 4·12-s − 12·13-s − 4·16-s + 6·18-s + 24·26-s + 14·27-s − 16·31-s + 8·32-s − 6·36-s − 4·37-s + 24·39-s + 14·41-s + 8·43-s + 8·48-s + 10·49-s − 24·52-s + 8·53-s − 28·54-s + 32·62-s − 8·64-s + 6·67-s + 4·71-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s − 9-s − 1.15·12-s − 3.32·13-s − 16-s + 1.41·18-s + 4.70·26-s + 2.69·27-s − 2.87·31-s + 1.41·32-s − 36-s − 0.657·37-s + 3.84·39-s + 2.18·41-s + 1.21·43-s + 1.15·48-s + 10/7·49-s − 3.32·52-s + 1.09·53-s − 3.81·54-s + 4.06·62-s − 64-s + 0.733·67-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1431954449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1431954449\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−12.61062107199992433816098545188, −11.90574367695315294970223748131, −11.79337993750628241072838176764, −10.95419872043741811311792184935, −10.82542994965107741918381756425, −10.34238760589740747154838606009, −9.674559408012646997019878510460, −9.124886471710456650569123649093, −9.112683266333877339492625096171, −8.139976705299177940483319246633, −7.65108178694573654080356757668, −7.07472717006458135359983507268, −6.93925357222468236882640820861, −5.70517991737646303550505789395, −5.54044890214768876050045668747, −4.92088943150135517246465238883, −4.14501128395594049818567157482, −2.68801915390584029317465905519, −2.26946229407000757273679838784, −0.41969665064217815584846674870,
0.41969665064217815584846674870, 2.26946229407000757273679838784, 2.68801915390584029317465905519, 4.14501128395594049818567157482, 4.92088943150135517246465238883, 5.54044890214768876050045668747, 5.70517991737646303550505789395, 6.93925357222468236882640820861, 7.07472717006458135359983507268, 7.65108178694573654080356757668, 8.139976705299177940483319246633, 9.112683266333877339492625096171, 9.124886471710456650569123649093, 9.674559408012646997019878510460, 10.34238760589740747154838606009, 10.82542994965107741918381756425, 10.95419872043741811311792184935, 11.79337993750628241072838176764, 11.90574367695315294970223748131, 12.61062107199992433816098545188
