| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3·7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 3·14-s − 15-s + 16-s + 17-s + 18-s − 8·19-s + 20-s − 3·21-s − 22-s + 3·23-s − 24-s + 25-s − 27-s + 3·28-s + 3·29-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 + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.654·21-s − 0.213·22-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.566·28-s + 0.557·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.330045470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.330045470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.107684214040800594956789165362, −7.28675564437691234121496846307, −6.49533457571750897353337508591, −5.94817536914287633839650924803, −5.15235385876827060898075186612, −4.62053010412022343272265656110, −3.97937321558304460942776715848, −2.68814432172962699810350152873, −1.99484733768626558303665074668, −0.933092633532134841021655551304,
0.933092633532134841021655551304, 1.99484733768626558303665074668, 2.68814432172962699810350152873, 3.97937321558304460942776715848, 4.62053010412022343272265656110, 5.15235385876827060898075186612, 5.94817536914287633839650924803, 6.49533457571750897353337508591, 7.28675564437691234121496846307, 8.107684214040800594956789165362
