| L(s) = 1 | + 16·2-s − 23.8·3-s + 256·4-s + 1.08e3·5-s − 381.·6-s + 4.09e3·8-s − 1.91e4·9-s + 1.73e4·10-s + 7.49e4·11-s − 6.10e3·12-s − 5.00e4·13-s − 2.58e4·15-s + 6.55e4·16-s + 1.18e5·17-s − 3.05e5·18-s + 8.04e5·19-s + 2.77e5·20-s + 1.19e6·22-s + 2.68e5·23-s − 9.76e4·24-s − 7.81e5·25-s − 8.00e5·26-s + 9.25e5·27-s − 5.29e6·29-s − 4.12e5·30-s + 5.46e6·31-s + 1.04e6·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.169·3-s + 0.5·4-s + 0.774·5-s − 0.120·6-s + 0.353·8-s − 0.971·9-s + 0.547·10-s + 1.54·11-s − 0.0849·12-s − 0.486·13-s − 0.131·15-s + 0.250·16-s + 0.345·17-s − 0.686·18-s + 1.41·19-s + 0.387·20-s + 1.09·22-s + 0.200·23-s − 0.0600·24-s − 0.400·25-s − 0.343·26-s + 0.335·27-s − 1.39·29-s − 0.0930·30-s + 1.06·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.842402831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.842402831\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 23.8T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.08e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 7.49e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.00e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.18e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.68e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.29e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.46e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.10e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.07e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.76e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.01e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.71e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.83e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.30e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.73e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.68e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.64e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.13e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.50e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.17e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.32e9T + 7.60e17T^{2} \) |
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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
−11.95950982188292969980054906563, −11.44463577430227305514080662059, −9.942734325431495800212319528480, −9.005291518707416478488557497644, −7.37765865383308536480519101541, −6.12364860101018010184990953955, −5.36526008007303834343053526813, −3.83432112969645885649029679411, −2.49703505890245532054508902291, −1.06229416323453628832462140900,
1.06229416323453628832462140900, 2.49703505890245532054508902291, 3.83432112969645885649029679411, 5.36526008007303834343053526813, 6.12364860101018010184990953955, 7.37765865383308536480519101541, 9.005291518707416478488557497644, 9.942734325431495800212319528480, 11.44463577430227305514080662059, 11.95950982188292969980054906563
