| L(s) = 1 | − 2.21·2-s − 3.09·4-s − 10.2·5-s − 27.8·7-s + 24.5·8-s + 22.7·10-s − 21.8·11-s − 22.8·13-s + 61.6·14-s − 29.7·16-s + 90.1·17-s − 24.5·19-s + 31.6·20-s + 48.4·22-s − 70.4·23-s − 19.9·25-s + 50.6·26-s + 86.0·28-s − 198.·29-s − 94.0·31-s − 130.·32-s − 199.·34-s + 285.·35-s + 102.·37-s + 54.4·38-s − 251.·40-s + 143.·41-s + ⋯ |
| L(s) = 1 | − 0.783·2-s − 0.386·4-s − 0.916·5-s − 1.50·7-s + 1.08·8-s + 0.717·10-s − 0.599·11-s − 0.487·13-s + 1.17·14-s − 0.464·16-s + 1.28·17-s − 0.296·19-s + 0.354·20-s + 0.469·22-s − 0.638·23-s − 0.159·25-s + 0.381·26-s + 0.581·28-s − 1.27·29-s − 0.544·31-s − 0.722·32-s − 1.00·34-s + 1.37·35-s + 0.456·37-s + 0.232·38-s − 0.995·40-s + 0.545·41-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 | \( 1 + 2.21T + 8T^{2} \) |
| 5 | \( 1 + 10.2T + 125T^{2} \) |
| 7 | \( 1 + 27.8T + 343T^{2} \) |
| 11 | \( 1 + 21.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 90.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 70.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 102.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 143.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 82.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 576.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 151.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 79.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 642.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 670.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 304.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 670.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 871.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 404.T + 9.12e5T^{2} \) |
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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
−8.267738238477571561906881333020, −7.65660686431698915080455770483, −7.16460338286911125211751389051, −6.00214669420352609030297424429, −5.18943038155036279066698217804, −3.98159933741092547828806604943, −3.52893361133460491066180902315, −2.29165477641337188302420698971, −0.71798000317903826017794350760, 0,
0.71798000317903826017794350760, 2.29165477641337188302420698971, 3.52893361133460491066180902315, 3.98159933741092547828806604943, 5.18943038155036279066698217804, 6.00214669420352609030297424429, 7.16460338286911125211751389051, 7.65660686431698915080455770483, 8.267738238477571561906881333020
