| L(s) = 1 | − 3·7-s + 3·11-s + 13-s − 7·17-s + 2·19-s − 23-s + 6·29-s + 10·31-s + 37-s − 41-s − 4·43-s + 4·47-s + 2·49-s + 53-s + 13·61-s − 8·67-s − 5·71-s + 10·73-s − 9·77-s + 79-s + 6·83-s + 9·89-s − 3·91-s − 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.904·11-s + 0.277·13-s − 1.69·17-s + 0.458·19-s − 0.208·23-s + 1.11·29-s + 1.79·31-s + 0.164·37-s − 0.156·41-s − 0.609·43-s + 0.583·47-s + 2/7·49-s + 0.137·53-s + 1.66·61-s − 0.977·67-s − 0.593·71-s + 1.17·73-s − 1.02·77-s + 0.112·79-s + 0.658·83-s + 0.953·89-s − 0.314·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.034893455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034893455\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.69304399616058, −13.34887125982608, −12.99846333448669, −12.26418457934160, −11.86827493125185, −11.50240083319593, −10.85869290629663, −10.17990642022797, −10.00131015676117, −9.178623307032249, −9.026439611984708, −8.366951133389259, −7.884108825558966, −6.938858733472144, −6.721446582067425, −6.300671093138383, −5.829042985578244, −4.880555802923170, −4.499776364965235, −3.840426761437968, −3.319053830882118, −2.651703419845765, −2.102609447626311, −1.133795507370208, −0.5012475561467960,
0.5012475561467960, 1.133795507370208, 2.102609447626311, 2.651703419845765, 3.319053830882118, 3.840426761437968, 4.499776364965235, 4.880555802923170, 5.829042985578244, 6.300671093138383, 6.721446582067425, 6.938858733472144, 7.884108825558966, 8.366951133389259, 9.026439611984708, 9.178623307032249, 10.00131015676117, 10.17990642022797, 10.85869290629663, 11.50240083319593, 11.86827493125185, 12.26418457934160, 12.99846333448669, 13.34887125982608, 13.69304399616058
