| L(s) = 1 | + 2-s − 3-s + 4-s + 1.35·5-s − 6-s − 7-s + 8-s + 9-s + 1.35·10-s − 4.29·11-s − 12-s − 14-s − 1.35·15-s + 16-s + 2.66·17-s + 18-s − 4.29·19-s + 1.35·20-s + 21-s − 4.29·22-s − 4.63·23-s − 24-s − 3.15·25-s − 27-s − 28-s + 5.54·29-s − 1.35·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.606·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.429·10-s − 1.29·11-s − 0.288·12-s − 0.267·14-s − 0.350·15-s + 0.250·16-s + 0.646·17-s + 0.235·18-s − 0.985·19-s + 0.303·20-s + 0.218·21-s − 0.915·22-s − 0.965·23-s − 0.204·24-s − 0.631·25-s − 0.192·27-s − 0.188·28-s + 1.02·29-s − 0.247·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.400631179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.400631179\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 19 | \( 1 + 4.29T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 - 5.51T + 31T^{2} \) |
| 37 | \( 1 - 3.53T + 37T^{2} \) |
| 41 | \( 1 - 9.45T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 8.20T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 + 4.87T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 + 5.62T + 83T^{2} \) |
| 89 | \( 1 + 9.81T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−7.88730280475831362304196155958, −7.04543724780516734833404379745, −6.20834011992144226178335123324, −5.91868047272880425165128077794, −5.16264770486478922947440877943, −4.49454858810118512447501815079, −3.66938354277451455100568008582, −2.63412286386940625106858155014, −2.08178636012757952688851218408, −0.70419552267290399117817170350,
0.70419552267290399117817170350, 2.08178636012757952688851218408, 2.63412286386940625106858155014, 3.66938354277451455100568008582, 4.49454858810118512447501815079, 5.16264770486478922947440877943, 5.91868047272880425165128077794, 6.20834011992144226178335123324, 7.04543724780516734833404379745, 7.88730280475831362304196155958
