| L(s) = 1 | − 2·3-s + 7-s + 9-s + 6·13-s + 4·19-s − 2·21-s − 6·25-s + 4·27-s − 8·31-s − 12·37-s − 12·39-s + 12·43-s + 49-s − 8·57-s − 10·61-s + 63-s + 12·67-s − 14·73-s + 12·75-s + 12·79-s − 11·81-s + 6·91-s + 16·93-s − 22·97-s − 8·103-s − 32·109-s + 24·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.917·19-s − 0.436·21-s − 6/5·25-s + 0.769·27-s − 1.43·31-s − 1.97·37-s − 1.92·39-s + 1.82·43-s + 1/7·49-s − 1.05·57-s − 1.28·61-s + 0.125·63-s + 1.46·67-s − 1.63·73-s + 1.38·75-s + 1.35·79-s − 1.22·81-s + 0.628·91-s + 1.65·93-s − 2.23·97-s − 0.788·103-s − 3.06·109-s + 2.27·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.084464597237824308330593676076, −7.49585969216145825404784041811, −7.12979176377730470228817324114, −6.57415454433544345972921728032, −6.17482252955815539017439801655, −5.64246920131951156149989739371, −5.45105814322366189846136783079, −5.04974234133287565966973305587, −4.21429763647040325331141519095, −3.86882454529629759126619122769, −3.36349234462471542459491489035, −2.58174753985468517343724252668, −1.65191646716236687209217935796, −1.16199732726620328775314891597, 0,
1.16199732726620328775314891597, 1.65191646716236687209217935796, 2.58174753985468517343724252668, 3.36349234462471542459491489035, 3.86882454529629759126619122769, 4.21429763647040325331141519095, 5.04974234133287565966973305587, 5.45105814322366189846136783079, 5.64246920131951156149989739371, 6.17482252955815539017439801655, 6.57415454433544345972921728032, 7.12979176377730470228817324114, 7.49585969216145825404784041811, 8.084464597237824308330593676076
