| L(s) = 1 | + 3·3-s + 30.4·7-s + 9·9-s − 31.4·11-s + 60.7·13-s + 121.·17-s + 14.4·19-s + 91.3·21-s + 13.6·23-s + 27·27-s − 76.0·29-s − 183.·31-s − 94.3·33-s + 37.3·37-s + 182.·39-s − 30.6·41-s + 327.·43-s − 449.·47-s + 583.·49-s + 363.·51-s + 301.·53-s + 43.3·57-s − 340.·59-s + 619.·61-s + 273.·63-s + 256.·67-s + 41.0·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.64·7-s + 0.333·9-s − 0.861·11-s + 1.29·13-s + 1.72·17-s + 0.174·19-s + 0.948·21-s + 0.124·23-s + 0.192·27-s − 0.486·29-s − 1.06·31-s − 0.497·33-s + 0.166·37-s + 0.748·39-s − 0.116·41-s + 1.16·43-s − 1.39·47-s + 1.70·49-s + 0.998·51-s + 0.782·53-s + 0.100·57-s − 0.752·59-s + 1.29·61-s + 0.547·63-s + 0.468·67-s + 0.0716·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.696683662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.696683662\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 30.4T + 343T^{2} \) |
| 11 | \( 1 + 31.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 327.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 449.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 340.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 619.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 256.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 499.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 19.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 257.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 914.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 521T + 9.12e5T^{2} \) |
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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
−9.250003417465389181431115914679, −8.321524299443468845432299014270, −7.942621541262771580872474297967, −7.21772166971784037980223097094, −5.76040464315429811051007289004, −5.19717380730770229118062057892, −4.09416959679725495460229601427, −3.16698256335793052501414345279, −1.90866995205237504424263989818, −1.05488863977570994093680552935,
1.05488863977570994093680552935, 1.90866995205237504424263989818, 3.16698256335793052501414345279, 4.09416959679725495460229601427, 5.19717380730770229118062057892, 5.76040464315429811051007289004, 7.21772166971784037980223097094, 7.942621541262771580872474297967, 8.321524299443468845432299014270, 9.250003417465389181431115914679
