L(s) = 1 | − 2.41·2-s + 1.84·3-s + 3.83·4-s − 5-s − 4.44·6-s − 7-s − 4.43·8-s + 0.392·9-s + 2.41·10-s + 5.68·11-s + 7.06·12-s − 3.60·13-s + 2.41·14-s − 1.84·15-s + 3.04·16-s + 3.68·17-s − 0.947·18-s − 1.18·19-s − 3.83·20-s − 1.84·21-s − 13.7·22-s − 1.47·23-s − 8.17·24-s + 25-s + 8.69·26-s − 4.80·27-s − 3.83·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.06·3-s + 1.91·4-s − 0.447·5-s − 1.81·6-s − 0.377·7-s − 1.56·8-s + 0.130·9-s + 0.763·10-s + 1.71·11-s + 2.03·12-s − 0.998·13-s + 0.645·14-s − 0.475·15-s + 0.761·16-s + 0.893·17-s − 0.223·18-s − 0.270·19-s − 0.857·20-s − 0.401·21-s − 2.92·22-s − 0.307·23-s − 1.66·24-s + 0.200·25-s + 1.70·26-s − 0.924·27-s − 0.725·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.003225935\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003225935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 1.84T + 3T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 + 1.18T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 - 0.109T + 29T^{2} \) |
| 31 | \( 1 + 0.502T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 4.46T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 + 2.03T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 7.76T + 89T^{2} \) |
| 97 | \( 1 - 19.1T + 97T^{2} \) |
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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
−7.917637708235917607782013232929, −7.47792316738663487344463470082, −6.81168479869495434925051501774, −6.22950626614977950300638224212, −5.04697878083210055989066138727, −3.84256769971314552829510109708, −3.36852442223154522108322099216, −2.37524622477527297015043365901, −1.68727288736396102421736103454, −0.60897750856675531588767544675,
0.60897750856675531588767544675, 1.68727288736396102421736103454, 2.37524622477527297015043365901, 3.36852442223154522108322099216, 3.84256769971314552829510109708, 5.04697878083210055989066138727, 6.22950626614977950300638224212, 6.81168479869495434925051501774, 7.47792316738663487344463470082, 7.917637708235917607782013232929