| L(s) = 1 | + 1.80·2-s + 1.24·4-s + 2·5-s − 1.35·8-s + 3.60·10-s − 1.80·11-s − 4.49·13-s − 4.93·16-s − 2.71·17-s − 6·19-s + 2.49·20-s − 3.24·22-s − 5.82·23-s − 25-s − 8.09·26-s − 1.55·29-s + 6.59·31-s − 6.18·32-s − 4.89·34-s − 11.1·37-s − 10.8·38-s − 2.71·40-s + 6.31·41-s − 0.576·43-s − 2.24·44-s − 10.5·46-s + 2.61·47-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.623·4-s + 0.894·5-s − 0.479·8-s + 1.13·10-s − 0.543·11-s − 1.24·13-s − 1.23·16-s − 0.658·17-s − 1.37·19-s + 0.557·20-s − 0.692·22-s − 1.21·23-s − 0.200·25-s − 1.58·26-s − 0.288·29-s + 1.18·31-s − 1.09·32-s − 0.838·34-s − 1.83·37-s − 1.75·38-s − 0.429·40-s + 0.986·41-s − 0.0879·43-s − 0.338·44-s − 1.54·46-s + 0.381·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 6.59T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 + 0.576T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 - 4.63T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 1.95T + 67T^{2} \) |
| 71 | \( 1 + 0.478T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 0.850T + 79T^{2} \) |
| 83 | \( 1 - 1.90T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 5.65T + 97T^{2} \) |
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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
−8.390120989498286185401076721750, −7.31567183391396743379966869897, −6.51174722408501183766724741303, −5.89320050740295061333001514707, −5.20056470209423979851985257923, −4.50767001848762291626256690473, −3.75770529070380217346444127638, −2.44595345389218832571345731479, −2.18458099564218129198177400467, 0,
2.18458099564218129198177400467, 2.44595345389218832571345731479, 3.75770529070380217346444127638, 4.50767001848762291626256690473, 5.20056470209423979851985257923, 5.89320050740295061333001514707, 6.51174722408501183766724741303, 7.31567183391396743379966869897, 8.390120989498286185401076721750
