| L(s) = 1 | + 3.69·2-s + 5.63·4-s − 18.7·5-s − 24.1·7-s − 8.74·8-s − 69.2·10-s − 49.2·11-s − 89.1·14-s − 77.3·16-s − 65.3·17-s + 109.·19-s − 105.·20-s − 181.·22-s − 83.2·23-s + 226.·25-s − 135.·28-s + 4.99·29-s − 255.·31-s − 215.·32-s − 241.·34-s + 452.·35-s + 93.6·37-s + 405.·38-s + 164.·40-s + 67.9·41-s + 142.·43-s − 277.·44-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.703·4-s − 1.67·5-s − 1.30·7-s − 0.386·8-s − 2.19·10-s − 1.35·11-s − 1.70·14-s − 1.20·16-s − 0.931·17-s + 1.32·19-s − 1.18·20-s − 1.76·22-s − 0.755·23-s + 1.81·25-s − 0.917·28-s + 0.0319·29-s − 1.48·31-s − 1.19·32-s − 1.21·34-s + 2.18·35-s + 0.416·37-s + 1.73·38-s + 0.648·40-s + 0.258·41-s + 0.505·43-s − 0.950·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5215884120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5215884120\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.69T + 8T^{2} \) |
| 5 | \( 1 + 18.7T + 125T^{2} \) |
| 7 | \( 1 + 24.1T + 343T^{2} \) |
| 11 | \( 1 + 49.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 65.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 83.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 4.99T + 2.43e4T^{2} \) |
| 31 | \( 1 + 255.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 93.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 67.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 379.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 389.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 133.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 620.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 119.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 361.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 748.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 514.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 260.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 833.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.080580770454259229818811273438, −8.073670789762516423060900792776, −7.35371119038961165903006033176, −6.59602884697031724928617763823, −5.60875431289262710762753314062, −4.82079416262484598825411546270, −3.90283862392021241131554161187, −3.35270152792110552979633087080, −2.58855974955339248906487103834, −0.26650693767055300953773789971,
0.26650693767055300953773789971, 2.58855974955339248906487103834, 3.35270152792110552979633087080, 3.90283862392021241131554161187, 4.82079416262484598825411546270, 5.60875431289262710762753314062, 6.59602884697031724928617763823, 7.35371119038961165903006033176, 8.073670789762516423060900792776, 9.080580770454259229818811273438
