| L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·13-s − 2·21-s − 2·25-s − 27-s + 2·31-s + 4·39-s − 12·43-s − 2·49-s + 2·63-s + 4·67-s − 4·73-s + 2·75-s + 2·79-s + 81-s − 8·91-s − 2·93-s − 4·97-s + 14·103-s − 28·109-s − 4·117-s − 10·121-s + 127-s + 12·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.10·13-s − 0.436·21-s − 2/5·25-s − 0.192·27-s + 0.359·31-s + 0.640·39-s − 1.82·43-s − 2/7·49-s + 0.251·63-s + 0.488·67-s − 0.468·73-s + 0.230·75-s + 0.225·79-s + 1/9·81-s − 0.838·91-s − 0.207·93-s − 0.406·97-s + 1.37·103-s − 2.68·109-s − 0.369·117-s − 0.909·121-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.355228056719484220618413667277, −7.915042549322017552013294512017, −7.34749345912496252778768508245, −7.10443759463298776482085485631, −6.44385497328251542195795428301, −6.10244522475230986004391950685, −5.38115732992980502370783565890, −5.02760889254603086368998282302, −4.69503515033856406763562590390, −4.06418244761568269037189629956, −3.43923698554145728159326366716, −2.66421386225630633678296667822, −2.00976852828776521752541979803, −1.27132147394671372709569079604, 0,
1.27132147394671372709569079604, 2.00976852828776521752541979803, 2.66421386225630633678296667822, 3.43923698554145728159326366716, 4.06418244761568269037189629956, 4.69503515033856406763562590390, 5.02760889254603086368998282302, 5.38115732992980502370783565890, 6.10244522475230986004391950685, 6.44385497328251542195795428301, 7.10443759463298776482085485631, 7.34749345912496252778768508245, 7.915042549322017552013294512017, 8.355228056719484220618413667277
