| L(s) = 1 | − 3·9-s + 3·11-s + 13-s − 7·17-s − 7·19-s + 6·23-s − 5·25-s − 5·29-s + 8·37-s − 2·43-s + 7·47-s − 3·53-s − 7·59-s + 7·61-s + 3·67-s + 5·71-s − 14·73-s + 6·79-s + 9·81-s + 14·97-s − 9·99-s + 101-s + 103-s + 107-s + 109-s + 113-s − 3·117-s + ⋯ |
| L(s) = 1 | − 9-s + 0.904·11-s + 0.277·13-s − 1.69·17-s − 1.60·19-s + 1.25·23-s − 25-s − 0.928·29-s + 1.31·37-s − 0.304·43-s + 1.02·47-s − 0.412·53-s − 0.911·59-s + 0.896·61-s + 0.366·67-s + 0.593·71-s − 1.63·73-s + 0.675·79-s + 81-s + 1.42·97-s − 0.904·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.277·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.364600538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364600538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.85386952097339, −16.03707147955444, −15.30582175869111, −14.94846291082759, −14.40951923868038, −13.71371395038598, −13.12926246626204, −12.73665626819115, −11.80928385705670, −11.22648999897744, −11.06148935520599, −10.23814192392204, −9.229433135872293, −8.992082488939664, −8.448392734186726, −7.649505853477506, −6.823673982778161, −6.264008121244548, −5.810429289767968, −4.762732276252866, −4.201691482073283, −3.466209782614132, −2.473632180928164, −1.866509757400316, −0.5292062225920924,
0.5292062225920924, 1.866509757400316, 2.473632180928164, 3.466209782614132, 4.201691482073283, 4.762732276252866, 5.810429289767968, 6.264008121244548, 6.823673982778161, 7.649505853477506, 8.448392734186726, 8.992082488939664, 9.229433135872293, 10.23814192392204, 11.06148935520599, 11.22648999897744, 11.80928385705670, 12.73665626819115, 13.12926246626204, 13.71371395038598, 14.40951923868038, 14.94846291082759, 15.30582175869111, 16.03707147955444, 16.85386952097339
