| L(s) = 1 | − 4.21·2-s + 9.76·4-s − 14.1·5-s − 29.9·7-s − 7.45·8-s + 59.5·10-s − 19.3·11-s − 60.3·13-s + 126.·14-s − 46.7·16-s + 11.7·17-s − 14.0·19-s − 137.·20-s + 81.5·22-s − 7.12·23-s + 74.6·25-s + 254.·26-s − 292.·28-s + 112.·29-s − 208.·31-s + 256.·32-s − 49.5·34-s + 422.·35-s − 51.3·37-s + 59.4·38-s + 105.·40-s − 233.·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.22·4-s − 1.26·5-s − 1.61·7-s − 0.329·8-s + 1.88·10-s − 0.530·11-s − 1.28·13-s + 2.40·14-s − 0.730·16-s + 0.167·17-s − 0.170·19-s − 1.54·20-s + 0.790·22-s − 0.0645·23-s + 0.596·25-s + 1.92·26-s − 1.97·28-s + 0.721·29-s − 1.20·31-s + 1.41·32-s − 0.249·34-s + 2.04·35-s − 0.228·37-s + 0.253·38-s + 0.416·40-s − 0.888·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.21T + 8T^{2} \) |
| 5 | \( 1 + 14.1T + 125T^{2} \) |
| 7 | \( 1 + 29.9T + 343T^{2} \) |
| 11 | \( 1 + 19.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.12T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 51.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 233.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 34.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 294.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 96.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 169.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 671.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 270.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 171.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 135.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 797.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 204.T + 9.12e5T^{2} \) |
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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
−8.450145204293625191394870203532, −7.50684569525686928552167658973, −7.23541809795365054442013145112, −6.41863106840297300630706205837, −5.17235497631929397387825694732, −4.06410729385574125597342035101, −3.15536917235073211202719477390, −2.23252819269832837841935675421, −0.60232608984522707346063179694, 0,
0.60232608984522707346063179694, 2.23252819269832837841935675421, 3.15536917235073211202719477390, 4.06410729385574125597342035101, 5.17235497631929397387825694732, 6.41863106840297300630706205837, 7.23541809795365054442013145112, 7.50684569525686928552167658973, 8.450145204293625191394870203532
