| L(s) = 1 | − 5-s + 2·11-s + 2·13-s + 4·17-s + 8·23-s + 25-s + 2·31-s + 8·37-s + 2·41-s + 2·43-s + 10·47-s + 2·53-s − 2·55-s + 4·59-s − 10·61-s − 2·65-s − 2·67-s − 12·71-s + 10·73-s − 16·79-s + 16·83-s − 4·85-s − 14·89-s + 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s + 0.554·13-s + 0.970·17-s + 1.66·23-s + 1/5·25-s + 0.359·31-s + 1.31·37-s + 0.312·41-s + 0.304·43-s + 1.45·47-s + 0.274·53-s − 0.269·55-s + 0.520·59-s − 1.28·61-s − 0.248·65-s − 0.244·67-s − 1.42·71-s + 1.17·73-s − 1.80·79-s + 1.75·83-s − 0.433·85-s − 1.48·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.945719159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.945719159\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.00164138655172, −14.37359850890480, −14.04097299981379, −13.24198207367703, −12.91258146053771, −12.24467657275233, −11.79563125311983, −11.23955169826699, −10.80148118024097, −10.18767518435292, −9.548567758717300, −8.949633834986394, −8.617849862467296, −7.807598066575872, −7.365250028278257, −6.851861030943796, −5.991442146287297, −5.742134413630214, −4.725994357334290, −4.385076949046866, −3.505417024711309, −3.115859826292636, −2.263137835081931, −1.184773792281958, −0.7633668472663102,
0.7633668472663102, 1.184773792281958, 2.263137835081931, 3.115859826292636, 3.505417024711309, 4.385076949046866, 4.725994357334290, 5.742134413630214, 5.991442146287297, 6.851861030943796, 7.365250028278257, 7.807598066575872, 8.617849862467296, 8.949633834986394, 9.548567758717300, 10.18767518435292, 10.80148118024097, 11.23955169826699, 11.79563125311983, 12.24467657275233, 12.91258146053771, 13.24198207367703, 14.04097299981379, 14.37359850890480, 15.00164138655172
