| L(s) = 1 | + 3-s − 2·19-s + 25-s − 27-s + 2·31-s + 2·37-s − 2·57-s + 75-s − 81-s + 2·93-s + 2·103-s − 2·109-s + 2·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 3-s − 2·19-s + 25-s − 27-s + 2·31-s + 2·37-s − 2·57-s + 75-s − 81-s + 2·93-s + 2·103-s − 2·109-s + 2·111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.682731761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682731761\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.347913553467959116379971356881, −8.829984375487463093238543542460, −8.620886641361845600039444231418, −8.191822468902443548097144435338, −8.059504669325826181063202993505, −7.51604790775848753725465989904, −7.18155410114895145859927380363, −6.51005196797831603293376521922, −6.28431738649007638057939817588, −6.12154568390454536581049972812, −5.37298834629383443464695512859, −4.92489926098771545132311834879, −4.44880349665391334457367252945, −4.12244356078066379432436476872, −3.73112614678026613213519520011, −2.85487618404965542030039888056, −2.84971420831768090795136390045, −2.29856121296980983520458619846, −1.71918953262364424806938255793, −0.845557552588317244927063311378,
0.845557552588317244927063311378, 1.71918953262364424806938255793, 2.29856121296980983520458619846, 2.84971420831768090795136390045, 2.85487618404965542030039888056, 3.73112614678026613213519520011, 4.12244356078066379432436476872, 4.44880349665391334457367252945, 4.92489926098771545132311834879, 5.37298834629383443464695512859, 6.12154568390454536581049972812, 6.28431738649007638057939817588, 6.51005196797831603293376521922, 7.18155410114895145859927380363, 7.51604790775848753725465989904, 8.059504669325826181063202993505, 8.191822468902443548097144435338, 8.620886641361845600039444231418, 8.829984375487463093238543542460, 9.347913553467959116379971356881
