| L(s) = 1 | + 4·11-s − 2·13-s + 2·17-s + 19-s − 4·23-s − 6·29-s − 4·31-s + 6·37-s − 10·41-s − 4·43-s + 12·47-s − 7·49-s + 6·53-s − 12·59-s − 2·61-s + 4·67-s + 8·71-s + 6·73-s + 4·79-s + 12·83-s − 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s − 0.834·23-s − 1.11·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s − 0.609·43-s + 1.75·47-s − 49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.450·79-s + 1.31·83-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.975655532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975655532\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.26106281404404, −13.65210538864199, −13.29236737722763, −12.46211191923838, −12.18104919073839, −11.77512163370166, −11.13232942908786, −10.73437220553920, −9.953801531973658, −9.598711560146575, −9.173055688800047, −8.598706661743850, −7.820852796315665, −7.620875595075042, −6.728898293640096, −6.535105106329721, −5.662828202249695, −5.335835468557689, −4.551468622625998, −3.901545955256834, −3.548580591806693, −2.735727427957525, −1.937613148541180, −1.421791280928730, −0.4712273274452571,
0.4712273274452571, 1.421791280928730, 1.937613148541180, 2.735727427957525, 3.548580591806693, 3.901545955256834, 4.551468622625998, 5.335835468557689, 5.662828202249695, 6.535105106329721, 6.728898293640096, 7.620875595075042, 7.820852796315665, 8.598706661743850, 9.173055688800047, 9.598711560146575, 9.953801531973658, 10.73437220553920, 11.13232942908786, 11.77512163370166, 12.18104919073839, 12.46211191923838, 13.29236737722763, 13.65210538864199, 14.26106281404404
