L(s) = 1 | + 2.73·2-s + 5.46·4-s + 3.73·5-s + 7-s + 9.46·8-s + 10.1·10-s + 0.732·13-s + 2.73·14-s + 14.9·16-s − 0.267·17-s − 8.19·19-s + 20.3·20-s − 6.73·23-s + 8.92·25-s + 2·26-s + 5.46·28-s + 4.73·29-s + 0.535·31-s + 21.8·32-s − 0.732·34-s + 3.73·35-s − 2.53·37-s − 22.3·38-s + 35.3·40-s + 4·41-s + 6.46·43-s − 18.3·46-s + ⋯ |
L(s) = 1 | + 1.93·2-s + 2.73·4-s + 1.66·5-s + 0.377·7-s + 3.34·8-s + 3.22·10-s + 0.203·13-s + 0.730·14-s + 3.73·16-s − 0.0649·17-s − 1.88·19-s + 4.55·20-s − 1.40·23-s + 1.78·25-s + 0.392·26-s + 1.03·28-s + 0.878·29-s + 0.0962·31-s + 3.86·32-s − 0.125·34-s + 0.630·35-s − 0.416·37-s − 3.63·38-s + 5.58·40-s + 0.624·41-s + 0.985·43-s − 2.71·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.17090793\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.17090793\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 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
−7.63736919324340261613472559676, −6.50602127941327054817591244639, −6.34576238036369101376869259730, −5.80486548497530681074414475484, −5.00712215030050945117590091581, −4.47735199066914661561583178891, −3.71158211495319005647780923355, −2.62279026696080044199097281026, −2.15534812477300282918118396614, −1.47879916872234585177897894189,
1.47879916872234585177897894189, 2.15534812477300282918118396614, 2.62279026696080044199097281026, 3.71158211495319005647780923355, 4.47735199066914661561583178891, 5.00712215030050945117590091581, 5.80486548497530681074414475484, 6.34576238036369101376869259730, 6.50602127941327054817591244639, 7.63736919324340261613472559676