L(s) = 1 | + 3-s + 8·5-s + 9-s + 8·15-s − 16·19-s + 8·23-s + 38·25-s + 27-s + 8·45-s + 8·47-s − 10·49-s − 16·57-s − 8·67-s + 8·69-s + 24·71-s − 20·73-s + 38·75-s + 81-s − 128·95-s + 20·97-s + 64·115-s − 6·121-s + 136·125-s + 127-s + 131-s + 8·135-s + 137-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 3.57·5-s + 1/3·9-s + 2.06·15-s − 3.67·19-s + 1.66·23-s + 38/5·25-s + 0.192·27-s + 1.19·45-s + 1.16·47-s − 1.42·49-s − 2.11·57-s − 0.977·67-s + 0.963·69-s + 2.84·71-s − 2.34·73-s + 4.38·75-s + 1/9·81-s − 13.1·95-s + 2.03·97-s + 5.96·115-s − 0.545·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.688·135-s + 0.0854·137-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
\(L(1)\) |
\(\approx\) |
\(4.697062509\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.697062509\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.743300071949026671666068984095, −8.551716675388505933005243425858, −7.69025257463999263217129645276, −6.89547356553480728554873620104, −6.52923673370585434887828168703, −6.39174188210860038385829426400, −5.83450410447989994046324048602, −5.46781891072947205774781799219, −4.69490327290429029526959386590, −4.59332442534898507060628341827, −3.49343359434470738584054327342, −2.63009703571906191631419054606, −2.35029669656051858459889649700, −1.91249343134731656186986110897, −1.28088417651653847704023663777,
1.28088417651653847704023663777, 1.91249343134731656186986110897, 2.35029669656051858459889649700, 2.63009703571906191631419054606, 3.49343359434470738584054327342, 4.59332442534898507060628341827, 4.69490327290429029526959386590, 5.46781891072947205774781799219, 5.83450410447989994046324048602, 6.39174188210860038385829426400, 6.52923673370585434887828168703, 6.89547356553480728554873620104, 7.69025257463999263217129645276, 8.551716675388505933005243425858, 8.743300071949026671666068984095