| L(s) = 1 | + 2.73·2-s − 3-s + 5.46·4-s − 3.73·5-s − 2.73·6-s − 7-s + 9.46·8-s + 9-s − 10.1·10-s − 5.46·12-s − 0.732·13-s − 2.73·14-s + 3.73·15-s + 14.9·16-s − 0.267·17-s + 2.73·18-s + 8.19·19-s − 20.3·20-s + 21-s + 6.73·23-s − 9.46·24-s + 8.92·25-s − 2·26-s − 27-s − 5.46·28-s + 4.73·29-s + 10.1·30-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.577·3-s + 2.73·4-s − 1.66·5-s − 1.11·6-s − 0.377·7-s + 3.34·8-s + 0.333·9-s − 3.22·10-s − 1.57·12-s − 0.203·13-s − 0.730·14-s + 0.963·15-s + 3.73·16-s − 0.0649·17-s + 0.643·18-s + 1.88·19-s − 4.55·20-s + 0.218·21-s + 1.40·23-s − 1.93·24-s + 1.78·25-s − 0.392·26-s − 0.192·27-s − 1.03·28-s + 0.878·29-s + 1.86·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.149738668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.149738668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 13 | \( 1 + 0.732T + 13T^{2} \) |
| 17 | \( 1 + 0.267T + 17T^{2} \) |
| 19 | \( 1 - 8.19T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 + 2.53T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 6.46T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 3.26T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 - 7.73T + 83T^{2} \) |
| 89 | \( 1 - 2.66T + 89T^{2} \) |
| 97 | \( 1 - 6.73T + 97T^{2} \) |
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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
−8.676281707668627280735190554943, −7.47460746358181216177985236969, −7.27285948416964649393150172779, −6.50983749682833170224018873803, −5.47305015773542285744535752276, −4.91083117771367685384378725219, −4.17499346732336562873704834000, −3.39604296156803352527871372482, −2.81954108000418080807248526963, −1.04011128561946551079017709498,
1.04011128561946551079017709498, 2.81954108000418080807248526963, 3.39604296156803352527871372482, 4.17499346732336562873704834000, 4.91083117771367685384378725219, 5.47305015773542285744535752276, 6.50983749682833170224018873803, 7.27285948416964649393150172779, 7.47460746358181216177985236969, 8.676281707668627280735190554943
