| L(s) = 1 | − 4·11-s − 6·13-s + 2·17-s − 4·19-s + 8·23-s − 6·29-s + 31-s + 2·37-s − 10·41-s − 4·43-s − 7·49-s − 10·53-s − 12·59-s − 2·61-s − 4·67-s − 2·73-s − 4·83-s + 14·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 1.66·13-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 1.11·29-s + 0.179·31-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s − 1.37·53-s − 1.56·59-s − 0.256·61-s − 0.488·67-s − 0.234·73-s − 0.439·83-s + 1.48·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.19949916839340, −13.50305627425079, −13.06597432424097, −12.76437208499301, −12.25337389323461, −11.76026035767977, −11.11269476570169, −10.71200445524117, −10.25642793427591, −9.694766074091709, −9.335534889084470, −8.710345603916990, −8.055860755394801, −7.698191808680919, −7.225506002242970, −6.674291782215578, −6.107180361845443, −5.320740869958003, −4.940782905197712, −4.699896956689135, −3.768914416525847, −3.009851292432664, −2.750984035364851, −1.969633493836992, −1.339242890695639, 0, 0,
1.339242890695639, 1.969633493836992, 2.750984035364851, 3.009851292432664, 3.768914416525847, 4.699896956689135, 4.940782905197712, 5.320740869958003, 6.107180361845443, 6.674291782215578, 7.225506002242970, 7.698191808680919, 8.055860755394801, 8.710345603916990, 9.335534889084470, 9.694766074091709, 10.25642793427591, 10.71200445524117, 11.11269476570169, 11.76026035767977, 12.25337389323461, 12.76437208499301, 13.06597432424097, 13.50305627425079, 14.19949916839340
