| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s − 2·13-s + 2·15-s − 2·17-s − 4·19-s + 21-s − 25-s + 27-s + 6·29-s + 8·31-s + 2·35-s + 10·37-s − 2·39-s − 2·41-s + 2·45-s + 12·47-s + 49-s − 2·51-s + 6·53-s − 4·57-s − 4·59-s + 14·61-s + 63-s − 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.338·35-s + 1.64·37-s − 0.320·39-s − 0.312·41-s + 0.298·45-s + 1.75·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 1.79·61-s + 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.622887571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.622887571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.33278354564165, −12.95008928986319, −12.30590003713509, −11.96102140536533, −11.32918952112344, −10.82351194417915, −10.22167384273259, −9.961895216500098, −9.501854598247495, −8.856739392783508, −8.531112999456958, −8.026463099530502, −7.499212862910907, −6.889655942171469, −6.384052018949784, −5.990893817161887, −5.322498956996774, −4.744541044594759, −4.270317114091067, −3.815874005582528, −2.775769525249102, −2.499564105825074, −2.081726624127602, −1.254757288741651, −0.6159850658115996,
0.6159850658115996, 1.254757288741651, 2.081726624127602, 2.499564105825074, 2.775769525249102, 3.815874005582528, 4.270317114091067, 4.744541044594759, 5.322498956996774, 5.990893817161887, 6.384052018949784, 6.889655942171469, 7.499212862910907, 8.026463099530502, 8.531112999456958, 8.856739392783508, 9.501854598247495, 9.961895216500098, 10.22167384273259, 10.82351194417915, 11.32918952112344, 11.96102140536533, 12.30590003713509, 12.95008928986319, 13.33278354564165
