| L(s) = 1 | + 2·5-s + 9-s − 4·13-s − 4·17-s + 3·25-s + 12·29-s − 4·37-s − 12·41-s + 2·45-s − 14·49-s + 12·53-s − 28·61-s − 8·65-s + 4·73-s + 81-s − 8·85-s + 4·89-s − 28·97-s + 28·101-s − 28·109-s + 12·113-s − 4·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s − 1.10·13-s − 0.970·17-s + 3/5·25-s + 2.22·29-s − 0.657·37-s − 1.87·41-s + 0.298·45-s − 2·49-s + 1.64·53-s − 3.58·61-s − 0.992·65-s + 0.468·73-s + 1/9·81-s − 0.867·85-s + 0.423·89-s − 2.84·97-s + 2.78·101-s − 2.68·109-s + 1.12·113-s − 0.369·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.985644465490789105614221217605, −7.46133055491293853299357806271, −6.84594392053537268258202260884, −6.73746543691429154932020719885, −6.25812475404630879226037514190, −5.73892987388308473685173648012, −5.10335741728591181213017059039, −4.66281454287229345249472136354, −4.59744237493557402533707286052, −3.64257092304627787883654309435, −3.01325599841700254495812174327, −2.56100826409250417816310835164, −1.90410956588849953091073290916, −1.30789464322825876552451590970, 0,
1.30789464322825876552451590970, 1.90410956588849953091073290916, 2.56100826409250417816310835164, 3.01325599841700254495812174327, 3.64257092304627787883654309435, 4.59744237493557402533707286052, 4.66281454287229345249472136354, 5.10335741728591181213017059039, 5.73892987388308473685173648012, 6.25812475404630879226037514190, 6.73746543691429154932020719885, 6.84594392053537268258202260884, 7.46133055491293853299357806271, 7.985644465490789105614221217605
