| L(s) = 1 | + 2·7-s − 13-s + 17-s + 2·19-s + 4·23-s − 6·29-s − 4·31-s + 10·37-s + 10·41-s − 10·43-s − 12·47-s − 3·49-s + 6·53-s − 6·59-s + 6·61-s + 4·67-s + 4·71-s − 2·73-s − 2·79-s − 12·83-s + 6·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.277·13-s + 0.242·17-s + 0.458·19-s + 0.834·23-s − 1.11·29-s − 0.718·31-s + 1.64·37-s + 1.56·41-s − 1.52·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.781·59-s + 0.768·61-s + 0.488·67-s + 0.474·71-s − 0.234·73-s − 0.225·79-s − 1.31·83-s + 0.635·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.253091060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.253091060\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.55902522274336, −11.81013017903556, −11.49885705589769, −11.09048204149914, −10.89855739513280, −10.00357917553626, −9.804383309645030, −9.315504974933325, −8.826126884892868, −8.300445551897152, −7.814037843303894, −7.527472526668109, −6.993355387284278, −6.479152198687153, −5.893765506863908, −5.406545054893565, −4.982028427604948, −4.566989904546233, −3.904509136631740, −3.474792013406520, −2.788677691389484, −2.352697854789831, −1.568315002606078, −1.241156337557487, −0.3871186258254792,
0.3871186258254792, 1.241156337557487, 1.568315002606078, 2.352697854789831, 2.788677691389484, 3.474792013406520, 3.904509136631740, 4.566989904546233, 4.982028427604948, 5.406545054893565, 5.893765506863908, 6.479152198687153, 6.993355387284278, 7.527472526668109, 7.814037843303894, 8.300445551897152, 8.826126884892868, 9.315504974933325, 9.804383309645030, 10.00357917553626, 10.89855739513280, 11.09048204149914, 11.49885705589769, 11.81013017903556, 12.55902522274336
