| L(s) = 1 | + 5-s + 11-s − 3·17-s + 4·19-s − 6·23-s + 25-s − 6·29-s + 7·31-s − 8·37-s − 8·41-s + 43-s − 2·47-s − 7·49-s + 12·53-s + 55-s − 3·59-s − 2·67-s + 8·71-s + 3·73-s − 8·79-s − 9·83-s − 3·85-s + 11·89-s + 4·95-s − 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.727·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s + 1.25·31-s − 1.31·37-s − 1.24·41-s + 0.152·43-s − 0.291·47-s − 49-s + 1.64·53-s + 0.134·55-s − 0.390·59-s − 0.244·67-s + 0.949·71-s + 0.351·73-s − 0.900·79-s − 0.987·83-s − 0.325·85-s + 1.16·89-s + 0.410·95-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.80922597784449, −12.26410625747539, −11.95705546019753, −11.45128187747887, −11.09588112932477, −10.43127130158893, −10.04456895473378, −9.697756287245539, −9.234164582074627, −8.587333231551258, −8.396668758875276, −7.718405920114710, −7.245517238938569, −6.698693998085282, −6.379950014974775, −5.762151681202975, −5.299535367227593, −4.901325808830011, −4.150358155883144, −3.828828735992520, −3.136378138430369, −2.659019525067246, −1.810860687719843, −1.688992322085307, −0.7400703033543668, 0,
0.7400703033543668, 1.688992322085307, 1.810860687719843, 2.659019525067246, 3.136378138430369, 3.828828735992520, 4.150358155883144, 4.901325808830011, 5.299535367227593, 5.762151681202975, 6.379950014974775, 6.698693998085282, 7.245517238938569, 7.718405920114710, 8.396668758875276, 8.587333231551258, 9.234164582074627, 9.697756287245539, 10.04456895473378, 10.43127130158893, 11.09588112932477, 11.45128187747887, 11.95705546019753, 12.26410625747539, 12.80922597784449
