| L(s) = 1 | − 5.27·2-s − 4.20·4-s + 35.7·5-s + 128.·7-s + 190.·8-s − 188.·10-s + 352.·13-s − 676.·14-s − 871.·16-s + 1.01e3·17-s + 2.19e3·19-s − 150.·20-s + 3.26e3·23-s − 1.84e3·25-s − 1.85e3·26-s − 539.·28-s + 3.38e3·29-s + 2.21e3·31-s − 1.51e3·32-s − 5.37e3·34-s + 4.58e3·35-s + 1.14e4·37-s − 1.15e4·38-s + 6.82e3·40-s − 3.17e3·41-s + 6.36e3·43-s − 1.72e4·46-s + ⋯ |
| L(s) = 1 | − 0.931·2-s − 0.131·4-s + 0.639·5-s + 0.989·7-s + 1.05·8-s − 0.595·10-s + 0.578·13-s − 0.922·14-s − 0.851·16-s + 0.855·17-s + 1.39·19-s − 0.0840·20-s + 1.28·23-s − 0.591·25-s − 0.539·26-s − 0.130·28-s + 0.747·29-s + 0.414·31-s − 0.261·32-s − 0.797·34-s + 0.632·35-s + 1.37·37-s − 1.29·38-s + 0.674·40-s − 0.295·41-s + 0.524·43-s − 1.20·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.219011532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219011532\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 5.27T + 32T^{2} \) |
| 5 | \( 1 - 35.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 128.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 352.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.19e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.26e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.38e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.38e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.92e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 931.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.90e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.36e4T + 8.58e9T^{2} \) |
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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
−9.239089138866041466553917403184, −8.304792751312176516543954475426, −7.79362569716987588396098996085, −6.88768836152834984998389030419, −5.61120001421188589900301307770, −5.01167619357000941007053153640, −3.92319837141881129268270894091, −2.57892806936196194347820670280, −1.30557698891431887502269955079, −0.907147598242951575835936548556,
0.907147598242951575835936548556, 1.30557698891431887502269955079, 2.57892806936196194347820670280, 3.92319837141881129268270894091, 5.01167619357000941007053153640, 5.61120001421188589900301307770, 6.88768836152834984998389030419, 7.79362569716987588396098996085, 8.304792751312176516543954475426, 9.239089138866041466553917403184
