| L(s) = 1 | + 3-s − 2·4-s − 5-s − 2·9-s − 2·12-s − 2·13-s − 15-s + 4·16-s + 2·20-s + 25-s − 5·27-s − 2·31-s + 4·36-s + 4·37-s − 2·39-s − 15·41-s − 2·43-s + 2·45-s + 4·48-s + 5·49-s + 4·52-s + 2·60-s − 8·64-s + 2·65-s − 17·67-s − 12·71-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s − 0.577·12-s − 0.554·13-s − 0.258·15-s + 16-s + 0.447·20-s + 1/5·25-s − 0.962·27-s − 0.359·31-s + 2/3·36-s + 0.657·37-s − 0.320·39-s − 2.34·41-s − 0.304·43-s + 0.298·45-s + 0.577·48-s + 5/7·49-s + 0.554·52-s + 0.258·60-s − 64-s + 0.248·65-s − 2.07·67-s − 1.42·71-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9226761759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9226761759\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
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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
−8.444706963898112735888714835795, −7.899730776002653045027032258182, −7.65831344851345385026799782283, −7.25448724059572333684990419978, −6.44282271377200560138917254194, −6.17853254219376064344222741839, −5.43306593168469972457616517364, −5.06459167277722007743156839489, −4.67507193113840755203474775842, −4.01440890241050124573434665069, −3.50368354438073832168882556807, −3.15054419083299668176277043762, −2.42220317359746828266257324843, −1.64273065393932961497618741239, −0.46818243583036888906067703050,
0.46818243583036888906067703050, 1.64273065393932961497618741239, 2.42220317359746828266257324843, 3.15054419083299668176277043762, 3.50368354438073832168882556807, 4.01440890241050124573434665069, 4.67507193113840755203474775842, 5.06459167277722007743156839489, 5.43306593168469972457616517364, 6.17853254219376064344222741839, 6.44282271377200560138917254194, 7.25448724059572333684990419978, 7.65831344851345385026799782283, 7.899730776002653045027032258182, 8.444706963898112735888714835795
