L(s) = 1 | − 2.73·2-s − 1.93·3-s + 5.45·4-s + 5.28·6-s + 7-s − 9.43·8-s + 0.750·9-s − 1.73·11-s − 10.5·12-s + 3.48·13-s − 2.73·14-s + 14.8·16-s + 6.76·17-s − 2.04·18-s + 3.21·19-s − 1.93·21-s + 4.72·22-s − 23-s + 18.2·24-s − 9.50·26-s + 4.35·27-s + 5.45·28-s + 0.961·29-s + 0.173·31-s − 21.6·32-s + 3.35·33-s − 18.4·34-s + ⋯ |
L(s) = 1 | − 1.93·2-s − 1.11·3-s + 2.72·4-s + 2.15·6-s + 0.377·7-s − 3.33·8-s + 0.250·9-s − 0.521·11-s − 3.04·12-s + 0.965·13-s − 0.729·14-s + 3.71·16-s + 1.64·17-s − 0.482·18-s + 0.736·19-s − 0.422·21-s + 1.00·22-s − 0.208·23-s + 3.73·24-s − 1.86·26-s + 0.838·27-s + 1.03·28-s + 0.178·29-s + 0.0310·31-s − 3.83·32-s + 0.583·33-s − 3.16·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5666111416\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5666111416\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 + 1.93T + 3T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 - 6.76T + 17T^{2} \) |
| 19 | \( 1 - 3.21T + 19T^{2} \) |
| 29 | \( 1 - 0.961T + 29T^{2} \) |
| 31 | \( 1 - 0.173T + 31T^{2} \) |
| 37 | \( 1 + 2.69T + 37T^{2} \) |
| 41 | \( 1 - 9.74T + 41T^{2} \) |
| 43 | \( 1 - 9.39T + 43T^{2} \) |
| 47 | \( 1 - 0.788T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 6.51T + 59T^{2} \) |
| 61 | \( 1 + 0.994T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 - 2.31T + 83T^{2} \) |
| 89 | \( 1 - 8.71T + 89T^{2} \) |
| 97 | \( 1 + 6.15T + 97T^{2} \) |
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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
−8.368548624214407651074961110794, −7.81505224618505415289716417101, −7.27990871491008235124559437153, −6.25612841617421503551930105002, −5.85784019086649406201948724984, −5.10100754134662403125755584775, −3.53641114235433799120637923652, −2.57209824070319848695998496206, −1.32187380789827544528057275638, −0.69529199769320432784270101249,
0.69529199769320432784270101249, 1.32187380789827544528057275638, 2.57209824070319848695998496206, 3.53641114235433799120637923652, 5.10100754134662403125755584775, 5.85784019086649406201948724984, 6.25612841617421503551930105002, 7.27990871491008235124559437153, 7.81505224618505415289716417101, 8.368548624214407651074961110794