L(s) = 1 | + 5-s + 5·11-s + 2·13-s − 17-s − 19-s − 4·23-s − 4·25-s + 6·29-s + 10·31-s − 11·43-s + 9·47-s − 10·53-s + 5·55-s + 4·59-s + 5·61-s + 2·65-s − 4·67-s − 8·71-s − 13·73-s + 4·79-s − 4·83-s − 85-s − 6·89-s − 95-s − 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.50·11-s + 0.554·13-s − 0.242·17-s − 0.229·19-s − 0.834·23-s − 4/5·25-s + 1.11·29-s + 1.79·31-s − 1.67·43-s + 1.31·47-s − 1.37·53-s + 0.674·55-s + 0.520·59-s + 0.640·61-s + 0.248·65-s − 0.488·67-s − 0.949·71-s − 1.52·73-s + 0.450·79-s − 0.439·83-s − 0.108·85-s − 0.635·89-s − 0.102·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.46133648462682, −13.82503250820964, −13.56470169807678, −13.11022012440381, −12.18213328944339, −11.94906733977036, −11.62516553773140, −10.85119779395483, −10.36784871283247, −9.753717494451968, −9.514255674441670, −8.706728090418644, −8.405292844044275, −7.879040263925871, −6.966137970902194, −6.592024518651686, −6.135682656476954, −5.685491697231906, −4.812657959704212, −4.237348943839264, −3.855509302721101, −3.032716423364294, −2.402042098909046, −1.525017860613316, −1.167100864658853, 0,
1.167100864658853, 1.525017860613316, 2.402042098909046, 3.032716423364294, 3.855509302721101, 4.237348943839264, 4.812657959704212, 5.685491697231906, 6.135682656476954, 6.592024518651686, 6.966137970902194, 7.879040263925871, 8.405292844044275, 8.706728090418644, 9.514255674441670, 9.753717494451968, 10.36784871283247, 10.85119779395483, 11.62516553773140, 11.94906733977036, 12.18213328944339, 13.11022012440381, 13.56470169807678, 13.82503250820964, 14.46133648462682