| L(s) = 1 | − 5.09·2-s + 17.9·4-s − 15.2·5-s − 21.0·7-s − 50.6·8-s + 77.7·10-s + 65.8·11-s − 72.1·13-s + 107.·14-s + 114.·16-s + 51.9·17-s + 10.1·19-s − 273.·20-s − 335.·22-s − 9.17·23-s + 108.·25-s + 367.·26-s − 378.·28-s + 32.6·29-s + 212.·31-s − 177.·32-s − 264.·34-s + 322.·35-s − 53.7·37-s − 51.6·38-s + 773.·40-s + 49.4·41-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 2.24·4-s − 1.36·5-s − 1.13·7-s − 2.23·8-s + 2.45·10-s + 1.80·11-s − 1.53·13-s + 2.05·14-s + 1.78·16-s + 0.740·17-s + 0.122·19-s − 3.06·20-s − 3.24·22-s − 0.0831·23-s + 0.864·25-s + 2.77·26-s − 2.55·28-s + 0.209·29-s + 1.23·31-s − 0.982·32-s − 1.33·34-s + 1.55·35-s − 0.238·37-s − 0.220·38-s + 3.05·40-s + 0.188·41-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
| \(L(2)\) |
\(\approx\) |
\(0.3224641318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3224641318\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.09T + 8T^{2} \) |
| 5 | \( 1 + 15.2T + 125T^{2} \) |
| 7 | \( 1 + 21.0T + 343T^{2} \) |
| 11 | \( 1 - 65.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.17T + 1.21e4T^{2} \) |
| 29 | \( 1 - 32.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 53.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 49.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 259.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 226.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 894.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 118.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 270.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 314.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 338.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 716.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 632.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 187.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 483.T + 9.12e5T^{2} \) |
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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
−8.763383764629791604475469166169, −8.040504597290977455233586893285, −7.27837406854261441561723206314, −6.86518746891966983445104441770, −6.08759966915170292223933527632, −4.51155129494431103233044929550, −3.51890405051621211116894395521, −2.71446421430331766148545475098, −1.31907311470229176465543334908, −0.36988235893859017013502553599,
0.36988235893859017013502553599, 1.31907311470229176465543334908, 2.71446421430331766148545475098, 3.51890405051621211116894395521, 4.51155129494431103233044929550, 6.08759966915170292223933527632, 6.86518746891966983445104441770, 7.27837406854261441561723206314, 8.040504597290977455233586893285, 8.763383764629791604475469166169
