| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s − 7-s − 3·8-s + 9-s + 10-s + 12-s + 6·13-s − 14-s − 15-s − 16-s − 2·17-s + 18-s + 8·19-s − 20-s + 21-s + 8·23-s + 3·24-s + 25-s + 6·26-s − 27-s + 28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.218·21-s + 1.66·23-s + 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176505 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176505 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.662870334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.662870334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.22523979262466, −12.86565444449467, −12.38958168042339, −11.74563980949719, −11.34482412860978, −11.03405614043072, −10.30796163719263, −9.903854926050562, −9.305508923421610, −9.011935701959123, −8.516737641899199, −7.902190678755556, −7.203242444884186, −6.630090399506445, −6.255345318290883, −5.783079752095317, −5.313201785103999, −4.783337100580988, −4.440183275103009, −3.505360479867428, −3.319483451242773, −2.787019988132620, −1.746487271249760, −1.011946019453785, −0.6325045442493565,
0.6325045442493565, 1.011946019453785, 1.746487271249760, 2.787019988132620, 3.319483451242773, 3.505360479867428, 4.440183275103009, 4.783337100580988, 5.313201785103999, 5.783079752095317, 6.255345318290883, 6.630090399506445, 7.203242444884186, 7.902190678755556, 8.516737641899199, 9.011935701959123, 9.305508923421610, 9.903854926050562, 10.30796163719263, 11.03405614043072, 11.34482412860978, 11.74563980949719, 12.38958168042339, 12.86565444449467, 13.22523979262466
