| L(s) = 1 | − 5·11-s − 5·13-s − 4·17-s + 2·19-s − 3·23-s + 10·29-s − 6·31-s + 5·37-s + 10·41-s − 10·43-s − 5·47-s − 7·49-s + 2·53-s − 5·59-s − 11·61-s + 5·71-s + 10·73-s − 12·79-s + 12·83-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.50·11-s − 1.38·13-s − 0.970·17-s + 0.458·19-s − 0.625·23-s + 1.85·29-s − 1.07·31-s + 0.821·37-s + 1.56·41-s − 1.52·43-s − 0.729·47-s − 49-s + 0.274·53-s − 0.650·59-s − 1.40·61-s + 0.593·71-s + 1.17·73-s − 1.35·79-s + 1.31·83-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9604271036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9604271036\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−16.36519587801333, −15.98835698454879, −15.45414596828312, −14.80979537400317, −14.31404596013256, −13.58822566432397, −13.11189420676802, −12.49181837022527, −12.03461412590077, −11.23251030655962, −10.71087525358442, −10.05514789331211, −9.614435071754631, −8.888677564193102, −7.906980897226176, −7.850331993942394, −6.928768492469941, −6.312462028479158, −5.433499535510152, −4.850429230945368, −4.372168957244054, −3.158084169635632, −2.604091976824446, −1.870876442145969, −0.4293034607542012,
0.4293034607542012, 1.870876442145969, 2.604091976824446, 3.158084169635632, 4.372168957244054, 4.850429230945368, 5.433499535510152, 6.312462028479158, 6.928768492469941, 7.850331993942394, 7.906980897226176, 8.888677564193102, 9.614435071754631, 10.05514789331211, 10.71087525358442, 11.23251030655962, 12.03461412590077, 12.49181837022527, 13.11189420676802, 13.58822566432397, 14.31404596013256, 14.80979537400317, 15.45414596828312, 15.98835698454879, 16.36519587801333
