| L(s) = 1 | − 2.61·2-s + 4.85·4-s − 7-s − 7.47·8-s + 5.47·11-s + 0.763·13-s + 2.61·14-s + 9.85·16-s + 7.70·17-s − 3.23·19-s − 14.3·22-s − 5·23-s − 2·26-s − 4.85·28-s − 4.70·29-s + 4.47·31-s − 10.8·32-s − 20.1·34-s + 5.47·37-s + 8.47·38-s + 8·41-s + 8.23·43-s + 26.5·44-s + 13.0·46-s − 7.23·47-s + 49-s + 3.70·52-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.42·4-s − 0.377·7-s − 2.64·8-s + 1.64·11-s + 0.211·13-s + 0.699·14-s + 2.46·16-s + 1.86·17-s − 0.742·19-s − 3.05·22-s − 1.04·23-s − 0.392·26-s − 0.917·28-s − 0.874·29-s + 0.803·31-s − 1.91·32-s − 3.46·34-s + 0.899·37-s + 1.37·38-s + 1.24·41-s + 1.25·43-s + 4.00·44-s + 1.93·46-s − 1.05·47-s + 0.142·49-s + 0.514·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7754087596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7754087596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 - 7.70T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 + 4.70T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 8.23T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 0.763T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 - 0.527T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 1.76T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 5.70T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.445926743058312038995117615854, −8.792025544568502473354343280102, −7.898763926742529603107324079522, −7.38422640039892577952146538504, −6.26172044559035611107708044191, −6.01101416446183916950254835053, −4.13465388679144480844576426172, −3.09296003911675803365689704610, −1.80571237854845747522753979069, −0.851799364962200221279542813047,
0.851799364962200221279542813047, 1.80571237854845747522753979069, 3.09296003911675803365689704610, 4.13465388679144480844576426172, 6.01101416446183916950254835053, 6.26172044559035611107708044191, 7.38422640039892577952146538504, 7.898763926742529603107324079522, 8.792025544568502473354343280102, 9.445926743058312038995117615854
