| 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
| \(L(1)\) |
\(\approx\) |
\(2.147580291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147580291\) |
| \(L(\frac{3}{2})\) |
|
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
−16.26133057729477, −15.93012595687163, −15.65166339057758, −14.83531736364611, −13.96618564591721, −13.91968448760137, −12.92664989618975, −12.67539434423559, −11.99224899276914, −11.07080651929219, −10.80180864515301, −10.22018241396305, −9.488782840561073, −8.781838721809350, −8.225795694649035, −7.585197286284415, −7.083562083528240, −5.978054318823097, −5.738551481913703, −4.896753149406847, −4.180813161563175, −3.159026705485288, −2.820262830989655, −1.588314286287165, −0.7123645069270895,
0.7123645069270895, 1.588314286287165, 2.820262830989655, 3.159026705485288, 4.180813161563175, 4.896753149406847, 5.738551481913703, 5.978054318823097, 7.083562083528240, 7.585197286284415, 8.225795694649035, 8.781838721809350, 9.488782840561073, 10.22018241396305, 10.80180864515301, 11.07080651929219, 11.99224899276914, 12.67539434423559, 12.92664989618975, 13.91968448760137, 13.96618564591721, 14.83531736364611, 15.65166339057758, 15.93012595687163, 16.26133057729477
