L(s) = 1 | − 2·3-s + 5-s + 9-s − 2·15-s − 4·23-s + 25-s + 4·27-s + 4·29-s − 4·43-s + 45-s + 4·47-s + 6·49-s − 12·67-s + 8·69-s − 16·71-s − 16·73-s − 2·75-s − 11·81-s − 8·87-s − 16·97-s + 4·101-s − 4·115-s + 10·121-s + 125-s + 127-s + 8·129-s + 131-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.516·15-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.742·29-s − 0.609·43-s + 0.149·45-s + 0.583·47-s + 6/7·49-s − 1.46·67-s + 0.963·69-s − 1.89·71-s − 1.87·73-s − 0.230·75-s − 1.22·81-s − 0.857·87-s − 1.62·97-s + 0.398·101-s − 0.373·115-s + 0.909·121-s + 0.0894·125-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.340550675439908834607126306333, −7.54776785489074273760473316510, −7.31810520828115958082961904738, −6.72952739658898814292199143838, −6.21392083232497598786968397515, −5.96339077192636481068384282336, −5.50821594029589561372147870726, −5.06358661139685213346031755584, −4.39400603172446289891117063197, −4.16977632743161296090171115808, −3.17306312543156255737439470129, −2.73950895106643991740122890990, −1.87248077085664228992419693977, −1.11318806735436084084095379142, 0,
1.11318806735436084084095379142, 1.87248077085664228992419693977, 2.73950895106643991740122890990, 3.17306312543156255737439470129, 4.16977632743161296090171115808, 4.39400603172446289891117063197, 5.06358661139685213346031755584, 5.50821594029589561372147870726, 5.96339077192636481068384282336, 6.21392083232497598786968397515, 6.72952739658898814292199143838, 7.31810520828115958082961904738, 7.54776785489074273760473316510, 8.340550675439908834607126306333