L(s) = 1 | − 2.66·2-s + 3-s + 5.12·4-s − 2.66·6-s + 0.454·7-s − 8.33·8-s + 9-s − 1.45·11-s + 5.12·12-s − 3.33·13-s − 1.21·14-s + 12.0·16-s − 2.66·18-s + 19-s + 0.454·21-s + 3.88·22-s − 6.79·23-s − 8.33·24-s + 8.90·26-s + 27-s + 2.33·28-s + 3·29-s + 8.79·31-s − 15.3·32-s − 1.45·33-s + 5.12·36-s + 10.2·37-s + ⋯ |
L(s) = 1 | − 1.88·2-s + 0.577·3-s + 2.56·4-s − 1.08·6-s + 0.171·7-s − 2.94·8-s + 0.333·9-s − 0.438·11-s + 1.47·12-s − 0.925·13-s − 0.324·14-s + 3.00·16-s − 0.629·18-s + 0.229·19-s + 0.0992·21-s + 0.827·22-s − 1.41·23-s − 1.70·24-s + 1.74·26-s + 0.192·27-s + 0.440·28-s + 0.557·29-s + 1.57·31-s − 2.71·32-s − 0.253·33-s + 0.853·36-s + 1.68·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8031940834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8031940834\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 7 | \( 1 - 0.454T + 7T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 6.79T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 8.79T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.97T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2.42T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 61 | \( 1 + 3.90T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 8.97T + 79T^{2} \) |
| 83 | \( 1 + 4.54T + 83T^{2} \) |
| 89 | \( 1 + 0.0901T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−9.609839927894325207105117949222, −8.712532779057635404647211517605, −7.981433715685128735233749861301, −7.60583346852054660816995680653, −6.68840310874937769841187580499, −5.77288139647343232522822230986, −4.32664668896782373582464097133, −2.78556194172172236322488934756, −2.21476187028307508214873968040, −0.817368586730479305433292227350,
0.817368586730479305433292227350, 2.21476187028307508214873968040, 2.78556194172172236322488934756, 4.32664668896782373582464097133, 5.77288139647343232522822230986, 6.68840310874937769841187580499, 7.60583346852054660816995680653, 7.981433715685128735233749861301, 8.712532779057635404647211517605, 9.609839927894325207105117949222