| L(s) = 1 | − 7-s + 4·9-s − 4·11-s − 4·23-s − 8·25-s + 6·29-s + 2·37-s − 4·43-s + 49-s − 18·53-s − 4·63-s + 4·77-s + 7·81-s − 16·99-s + 16·107-s + 2·109-s − 8·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + 163-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 4/3·9-s − 1.20·11-s − 0.834·23-s − 8/5·25-s + 1.11·29-s + 0.328·37-s − 0.609·43-s + 1/7·49-s − 2.47·53-s − 0.503·63-s + 0.455·77-s + 7/9·81-s − 1.60·99-s + 1.54·107-s + 0.191·109-s − 0.752·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.315·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
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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
−8.303226924304951148068643602269, −7.994534146963523785611654752610, −7.63023306509678470157517664433, −7.22524746427421794534272710215, −6.51326492882019781344329095505, −6.26740326938018085578137265544, −5.68790156814063662177322866328, −5.00913016673686805121560114503, −4.65087321035173271084630102919, −4.03526499565280780433318465825, −3.52611871923192252336070883269, −2.78718576412331827975228766905, −2.12451232148835626510068334478, −1.37891046296909050262754276481, 0,
1.37891046296909050262754276481, 2.12451232148835626510068334478, 2.78718576412331827975228766905, 3.52611871923192252336070883269, 4.03526499565280780433318465825, 4.65087321035173271084630102919, 5.00913016673686805121560114503, 5.68790156814063662177322866328, 6.26740326938018085578137265544, 6.51326492882019781344329095505, 7.22524746427421794534272710215, 7.63023306509678470157517664433, 7.994534146963523785611654752610, 8.303226924304951148068643602269
