L(s) = 1 | + 3-s + 3.23·5-s − 7-s + 9-s + 1.23·11-s + 13-s + 3.23·15-s + 6.47·17-s − 2.47·19-s − 21-s + 4.47·23-s + 5.47·25-s + 27-s − 0.472·29-s + 1.23·33-s − 3.23·35-s − 4.47·37-s + 39-s + 0.763·41-s − 4·43-s + 3.23·45-s + 5.70·47-s + 49-s + 6.47·51-s + 10·53-s + 4.00·55-s − 2.47·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.44·5-s − 0.377·7-s + 0.333·9-s + 0.372·11-s + 0.277·13-s + 0.835·15-s + 1.56·17-s − 0.567·19-s − 0.218·21-s + 0.932·23-s + 1.09·25-s + 0.192·27-s − 0.0876·29-s + 0.215·33-s − 0.546·35-s − 0.735·37-s + 0.160·39-s + 0.119·41-s − 0.609·43-s + 0.482·45-s + 0.832·47-s + 0.142·49-s + 0.906·51-s + 1.37·53-s + 0.539·55-s − 0.327·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.480337754\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.480337754\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.651649770145004115639482186650, −7.56922666396540582258393662329, −6.92254869144040969255708571461, −6.06343308363972308269791130046, −5.60722376617108531608534470735, −4.67878770293101891752719459066, −3.58481600481327241919422902819, −2.90112733457988000382521603221, −1.94948974977672129033936294066, −1.11124295676329064788685702149,
1.11124295676329064788685702149, 1.94948974977672129033936294066, 2.90112733457988000382521603221, 3.58481600481327241919422902819, 4.67878770293101891752719459066, 5.60722376617108531608534470735, 6.06343308363972308269791130046, 6.92254869144040969255708571461, 7.56922666396540582258393662329, 8.651649770145004115639482186650