| L(s) = 1 | − 1.83·2-s − 4.62·4-s − 9.40·5-s + 19.7·7-s + 23.2·8-s + 17.2·10-s − 23.4·11-s − 4.74·13-s − 36.3·14-s − 5.68·16-s + 90.5·17-s − 133.·19-s + 43.4·20-s + 43.1·22-s + 97.8·23-s − 36.5·25-s + 8.71·26-s − 91.3·28-s + 6.66·29-s − 70.0·31-s − 175.·32-s − 166.·34-s − 185.·35-s − 34.2·37-s + 245.·38-s − 218.·40-s + 135.·41-s + ⋯ |
| L(s) = 1 | − 0.649·2-s − 0.577·4-s − 0.841·5-s + 1.06·7-s + 1.02·8-s + 0.546·10-s − 0.642·11-s − 0.101·13-s − 0.693·14-s − 0.0888·16-s + 1.29·17-s − 1.61·19-s + 0.485·20-s + 0.417·22-s + 0.886·23-s − 0.292·25-s + 0.0657·26-s − 0.616·28-s + 0.0426·29-s − 0.405·31-s − 0.967·32-s − 0.839·34-s − 0.897·35-s − 0.152·37-s + 1.04·38-s − 0.862·40-s + 0.516·41-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 | \( 1 + 1.83T + 8T^{2} \) |
| 5 | \( 1 + 9.40T + 125T^{2} \) |
| 7 | \( 1 - 19.7T + 343T^{2} \) |
| 11 | \( 1 + 23.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.74T + 2.19e3T^{2} \) |
| 17 | \( 1 - 90.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6.66T + 2.43e4T^{2} \) |
| 31 | \( 1 + 70.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 34.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 43.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 87.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 486.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 610.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 101.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 162.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 534.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 337.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 120.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 290.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 801.T + 9.12e5T^{2} \) |
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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
−8.501245852688140242248399654939, −7.63180380950921307560646778705, −7.31761052882575457799427468036, −5.85324372805132388586755346083, −4.96396380368295167904207081304, −4.36892946554691516973336752363, −3.47490993764032411171141258866, −2.09765662200844985251549918105, −1.00511658508140116800508369025, 0,
1.00511658508140116800508369025, 2.09765662200844985251549918105, 3.47490993764032411171141258866, 4.36892946554691516973336752363, 4.96396380368295167904207081304, 5.85324372805132388586755346083, 7.31761052882575457799427468036, 7.63180380950921307560646778705, 8.501245852688140242248399654939
