| L(s) = 1 | − 5-s − 6·13-s + 2·17-s + 8·19-s + 8·23-s + 25-s − 2·29-s + 4·31-s + 2·37-s − 6·41-s + 4·43-s − 8·47-s + 10·53-s + 4·59-s − 2·61-s + 6·65-s + 4·67-s − 12·71-s + 2·73-s − 8·79-s − 4·83-s − 2·85-s − 6·89-s − 8·95-s + 18·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.66·13-s + 0.485·17-s + 1.83·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1.37·53-s + 0.520·59-s − 0.256·61-s + 0.744·65-s + 0.488·67-s − 1.42·71-s + 0.234·73-s − 0.900·79-s − 0.439·83-s − 0.216·85-s − 0.635·89-s − 0.820·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.347225134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.347225134\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.27304418040579, −12.95720278476649, −12.40866736848789, −11.81282884760039, −11.65152333587092, −11.18579542347637, −10.38377155965304, −9.949388408853945, −9.710144453552129, −8.969461932616944, −8.680358513366737, −7.794257862742752, −7.553502638246126, −7.101626996186816, −6.708607775588003, −5.782062858016991, −5.378321409363272, −4.802821531416204, −4.542105556206608, −3.579417507562331, −3.118383112623330, −2.717138935055495, −1.909997468105602, −1.060092750642424, −0.5257566409606752,
0.5257566409606752, 1.060092750642424, 1.909997468105602, 2.717138935055495, 3.118383112623330, 3.579417507562331, 4.542105556206608, 4.802821531416204, 5.378321409363272, 5.782062858016991, 6.708607775588003, 7.101626996186816, 7.553502638246126, 7.794257862742752, 8.680358513366737, 8.969461932616944, 9.710144453552129, 9.949388408853945, 10.38377155965304, 11.18579542347637, 11.65152333587092, 11.81282884760039, 12.40866736848789, 12.95720278476649, 13.27304418040579
