L(s) = 1 | − 2-s + 1.73·3-s + 4-s − 3.73·5-s − 1.73·6-s + 7-s − 8-s + 3.73·10-s − 1.46·11-s + 1.73·12-s − 14-s − 6.46·15-s + 16-s + 3.46·17-s + 3.46·19-s − 3.73·20-s + 1.73·21-s + 1.46·22-s + 1.53·23-s − 1.73·24-s + 8.92·25-s − 5.19·27-s + 28-s − 4.92·29-s + 6.46·30-s − 7.46·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.00·3-s + 0.5·4-s − 1.66·5-s − 0.707·6-s + 0.377·7-s − 0.353·8-s + 1.18·10-s − 0.441·11-s + 0.500·12-s − 0.267·14-s − 1.66·15-s + 0.250·16-s + 0.840·17-s + 0.794·19-s − 0.834·20-s + 0.377·21-s + 0.312·22-s + 0.320·23-s − 0.353·24-s + 1.78·25-s − 1.00·27-s + 0.188·28-s − 0.915·29-s + 1.18·30-s − 1.34·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.190780746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.190780746\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8.92T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 + 4.46T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.985482468481825860122979382925, −8.014670133378411148586388291917, −7.65111527987991316272985377897, −7.35849111478084463727392316474, −5.92066937541887510734909664510, −4.90396048904913994711295690737, −3.73706899370529008336117419723, −3.27772453528571475243015271567, −2.21925446904970875547353981392, −0.73042288384820641168491640541,
0.73042288384820641168491640541, 2.21925446904970875547353981392, 3.27772453528571475243015271567, 3.73706899370529008336117419723, 4.90396048904913994711295690737, 5.92066937541887510734909664510, 7.35849111478084463727392316474, 7.65111527987991316272985377897, 8.014670133378411148586388291917, 8.985482468481825860122979382925