| L(s) = 1 | + 29.1·2-s − 6.46·3-s + 334.·4-s + 2.25e3·5-s − 188.·6-s + 6.42e3·7-s − 5.15e3·8-s − 1.96e4·9-s + 6.56e4·10-s + 1.46e4·11-s − 2.16e3·12-s − 5.92e4·13-s + 1.87e5·14-s − 1.45e4·15-s − 3.21e5·16-s − 2.53e5·17-s − 5.71e5·18-s − 4.67e5·19-s + 7.55e5·20-s − 4.15e4·21-s + 4.26e5·22-s + 8.06e5·23-s + 3.33e4·24-s + 3.13e6·25-s − 1.72e6·26-s + 2.54e5·27-s + 2.15e6·28-s + ⋯ |
| L(s) = 1 | + 1.28·2-s − 0.0461·3-s + 0.654·4-s + 1.61·5-s − 0.0592·6-s + 1.01·7-s − 0.444·8-s − 0.997·9-s + 2.07·10-s + 0.301·11-s − 0.0301·12-s − 0.575·13-s + 1.30·14-s − 0.0744·15-s − 1.22·16-s − 0.736·17-s − 1.28·18-s − 0.822·19-s + 1.05·20-s − 0.0466·21-s + 0.387·22-s + 0.601·23-s + 0.0205·24-s + 1.60·25-s − 0.740·26-s + 0.0921·27-s + 0.661·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.257266025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.257266025\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 1.46e4T \) |
| good | 2 | \( 1 - 29.1T + 512T^{2} \) |
| 3 | \( 1 + 6.46T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.25e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.42e3T + 4.03e7T^{2} \) |
| 13 | \( 1 + 5.92e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.67e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 8.06e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.92e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.58e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.33e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.21e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.38e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.71e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.27e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.92e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.94e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.52e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.07e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.91e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.43e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.32e8T + 7.60e17T^{2} \) |
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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
−17.97163808397069882763492643202, −17.11831496746478397243156797083, −14.72278523750846249037422122689, −14.11037696070867584897598599490, −12.85013290448440737667865674696, −11.14779313431454588510599909669, −9.056127268712393059427275425257, −6.15498684781662113805250059740, −4.92026963276836876827561494236, −2.34113318749150711438918636882,
2.34113318749150711438918636882, 4.92026963276836876827561494236, 6.15498684781662113805250059740, 9.056127268712393059427275425257, 11.14779313431454588510599909669, 12.85013290448440737667865674696, 14.11037696070867584897598599490, 14.72278523750846249037422122689, 17.11831496746478397243156797083, 17.97163808397069882763492643202
