L(s) = 1 | − 3.11·2-s + 1.72·4-s + 0.660·5-s − 24.6·7-s + 19.5·8-s − 2.05·10-s − 27.1·11-s + 16.3·13-s + 76.8·14-s − 74.8·16-s − 14.7·17-s − 127.·19-s + 1.13·20-s + 84.7·22-s − 159.·23-s − 124.·25-s − 51.0·26-s − 42.5·28-s − 229.·29-s + 244.·31-s + 76.8·32-s + 45.9·34-s − 16.2·35-s + 49.5·37-s + 399.·38-s + 12.9·40-s + 219.·41-s + ⋯ |
L(s) = 1 | − 1.10·2-s + 0.215·4-s + 0.0590·5-s − 1.33·7-s + 0.864·8-s − 0.0651·10-s − 0.744·11-s + 0.349·13-s + 1.46·14-s − 1.16·16-s − 0.210·17-s − 1.54·19-s + 0.0127·20-s + 0.821·22-s − 1.44·23-s − 0.996·25-s − 0.384·26-s − 0.287·28-s − 1.46·29-s + 1.41·31-s + 0.424·32-s + 0.231·34-s − 0.0785·35-s + 0.220·37-s + 1.70·38-s + 0.0510·40-s + 0.835·41-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
\(L(2)\) |
\(\approx\) |
\(0.003305123053\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.003305123053\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.11T + 8T^{2} \) |
| 5 | \( 1 - 0.660T + 125T^{2} \) |
| 7 | \( 1 + 24.6T + 343T^{2} \) |
| 11 | \( 1 + 27.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 159.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 244.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 49.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 219.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 10.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 337.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 636.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 154.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 213.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 116.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 922.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 529.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 266.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.51e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 96.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−8.724664079483917523103613341033, −8.095966908940755883851956230753, −7.43349152193028624438814174437, −6.37674404001005895256798974312, −5.93269695104014748045753749063, −4.56068787539105775306305843331, −3.81060660295066022593081843748, −2.60962992457781004391324410394, −1.65404595073461485553014211065, −0.03007384974866785182331538260,
0.03007384974866785182331538260, 1.65404595073461485553014211065, 2.60962992457781004391324410394, 3.81060660295066022593081843748, 4.56068787539105775306305843331, 5.93269695104014748045753749063, 6.37674404001005895256798974312, 7.43349152193028624438814174437, 8.095966908940755883851956230753, 8.724664079483917523103613341033