| L(s) = 1 | + 1.73·2-s + 3-s + 1.99·4-s + 1.73·6-s − 7-s + 1.73·8-s + 9-s − 1.73·11-s + 1.99·12-s + 13-s − 1.73·14-s + 0.999·16-s + 1.73·17-s + 1.73·18-s + 19-s − 21-s − 2.99·22-s − 1.73·23-s + 1.73·24-s + 25-s + 1.73·26-s + 27-s − 1.99·28-s − 1.73·33-s + 2.99·34-s + 1.99·36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 3-s + 1.99·4-s + 1.73·6-s − 7-s + 1.73·8-s + 9-s − 1.73·11-s + 1.99·12-s + 13-s − 1.73·14-s + 0.999·16-s + 1.73·17-s + 1.73·18-s + 19-s − 21-s − 2.99·22-s − 1.73·23-s + 1.73·24-s + 25-s + 1.73·26-s + 27-s − 1.99·28-s − 1.73·33-s + 2.99·34-s + 1.99·36-s − 2·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.464187472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.464187472\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 - 1.73T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + 1.73T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 2T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
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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
−8.383416791477075678389881412363, −7.83791283385432719230758926576, −7.06099375282803402872561956323, −6.31469317192124308729689783003, −5.46231919568844320873171948898, −4.98811436218841216127193977015, −3.73176428947809208247619090546, −3.33756660365528340684895554425, −2.82705785723055175001199538647, −1.70787857830629723799123358032,
1.70787857830629723799123358032, 2.82705785723055175001199538647, 3.33756660365528340684895554425, 3.73176428947809208247619090546, 4.98811436218841216127193977015, 5.46231919568844320873171948898, 6.31469317192124308729689783003, 7.06099375282803402872561956323, 7.83791283385432719230758926576, 8.383416791477075678389881412363
