| L(s) = 1 | + 3-s + 9-s − 8·11-s − 4·13-s + 8·23-s + 6·25-s + 27-s − 8·33-s + 4·37-s − 4·39-s + 8·47-s − 10·49-s + 8·59-s − 28·61-s + 8·69-s + 24·71-s − 20·73-s + 6·75-s + 81-s + 24·83-s + 20·97-s − 8·99-s − 8·107-s + 12·109-s + 4·111-s − 4·117-s + 26·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 2.41·11-s − 1.10·13-s + 1.66·23-s + 6/5·25-s + 0.192·27-s − 1.39·33-s + 0.657·37-s − 0.640·39-s + 1.16·47-s − 1.42·49-s + 1.04·59-s − 3.58·61-s + 0.963·69-s + 2.84·71-s − 2.34·73-s + 0.692·75-s + 1/9·81-s + 2.63·83-s + 2.03·97-s − 0.804·99-s − 0.773·107-s + 1.14·109-s + 0.379·111-s − 0.369·117-s + 2.36·121-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
| \(L(1)\) |
\(\approx\) |
\(1.660662376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660662376\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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, −7.83023979799298730526640872377, −7.69025257463999263217129645276, −7.59855962266761916039826076180, −6.69313352344866875849027252270, −6.52923673370585434887828168703, −5.46781891072947205774781799219, −5.34421672308299510449169414652, −4.62659376615273221460753154913, −4.59332442534898507060628341827, −3.38713852975091260060256278122, −2.92466309841837341058256028171, −2.63009703571906191631419054606, −1.91249343134731656186986110897, −0.65344977115887373240163738777,
0.65344977115887373240163738777, 1.91249343134731656186986110897, 2.63009703571906191631419054606, 2.92466309841837341058256028171, 3.38713852975091260060256278122, 4.59332442534898507060628341827, 4.62659376615273221460753154913, 5.34421672308299510449169414652, 5.46781891072947205774781799219, 6.52923673370585434887828168703, 6.69313352344866875849027252270, 7.59855962266761916039826076180, 7.69025257463999263217129645276, 7.83023979799298730526640872377, 8.743300071949026671666068984095
