| L(s) = 1 | − 4.53·2-s + 12.5·4-s − 19.7·5-s + 34.6·7-s − 20.8·8-s + 89.4·10-s + 41.4·11-s − 54.5·13-s − 157.·14-s − 6.12·16-s + 60.1·17-s + 108.·19-s − 248.·20-s − 188.·22-s + 49.6·23-s + 263.·25-s + 247.·26-s + 436.·28-s − 118.·29-s − 118.·31-s + 194.·32-s − 272.·34-s − 683.·35-s + 90.9·37-s − 493.·38-s + 410.·40-s + 397.·41-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.57·4-s − 1.76·5-s + 1.87·7-s − 0.921·8-s + 2.82·10-s + 1.13·11-s − 1.16·13-s − 3.00·14-s − 0.0957·16-s + 0.857·17-s + 1.31·19-s − 2.77·20-s − 1.82·22-s + 0.449·23-s + 2.10·25-s + 1.86·26-s + 2.94·28-s − 0.761·29-s − 0.687·31-s + 1.07·32-s − 1.37·34-s − 3.29·35-s + 0.403·37-s − 2.10·38-s + 1.62·40-s + 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.047141630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047141630\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 4.53T + 8T^{2} \) |
| 5 | \( 1 + 19.7T + 125T^{2} \) |
| 7 | \( 1 - 34.6T + 343T^{2} \) |
| 11 | \( 1 - 41.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 118.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 90.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 397.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 418.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 371.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 396.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 139.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 636.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 222.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 99.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 600.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 99.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 80.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−8.757635919577280623297348776749, −7.80794673325885827906327187283, −7.44029514874505736023383344831, −7.26178867854967331349859547857, −5.54750428358497319553229663507, −4.56731255733630474489409432264, −3.87436813012362534470333038634, −2.50859577587776742445270009869, −1.25518672692722481609792433731, −0.71392300550174348723682857423,
0.71392300550174348723682857423, 1.25518672692722481609792433731, 2.50859577587776742445270009869, 3.87436813012362534470333038634, 4.56731255733630474489409432264, 5.54750428358497319553229663507, 7.26178867854967331349859547857, 7.44029514874505736023383344831, 7.80794673325885827906327187283, 8.757635919577280623297348776749
