| 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$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−8.282381410165521860675726776780, −7.899627208563166987227072300312, −7.47488834419725943796237155325, −7.04540176019289354331595439652, −6.61448125181127928261834183745, −6.06854381910577730789124541104, −5.52326095536142766971265625805, −5.07120668817404130845367980522, −4.32270848084311449043743480172, −4.01009346990188471605532841114, −3.29793890884212981557473009994, −2.70941766455383713630040989598, −2.29625620351786037369134469371, −1.32630931564126395352496472072, 0,
1.32630931564126395352496472072, 2.29625620351786037369134469371, 2.70941766455383713630040989598, 3.29793890884212981557473009994, 4.01009346990188471605532841114, 4.32270848084311449043743480172, 5.07120668817404130845367980522, 5.52326095536142766971265625805, 6.06854381910577730789124541104, 6.61448125181127928261834183745, 7.04540176019289354331595439652, 7.47488834419725943796237155325, 7.899627208563166987227072300312, 8.282381410165521860675726776780
