| L(s) = 1 | − 2.89·3-s − 16.6·7-s − 18.5·9-s − 19.1·11-s − 61.7·13-s − 30.3·17-s + 59.1·19-s + 48.4·21-s − 205.·23-s + 132.·27-s − 8.38·29-s − 331.·31-s + 55.6·33-s − 266.·37-s + 179.·39-s − 320.·41-s + 83.1·43-s + 276.·47-s − 64.2·49-s + 88.0·51-s − 390.·53-s − 171.·57-s + 779.·59-s + 483.·61-s + 310.·63-s + 123.·67-s + 596.·69-s + ⋯ |
| L(s) = 1 | − 0.557·3-s − 0.901·7-s − 0.688·9-s − 0.526·11-s − 1.31·13-s − 0.433·17-s + 0.714·19-s + 0.502·21-s − 1.86·23-s + 0.942·27-s − 0.0536·29-s − 1.91·31-s + 0.293·33-s − 1.18·37-s + 0.735·39-s − 1.22·41-s + 0.294·43-s + 0.857·47-s − 0.187·49-s + 0.241·51-s − 1.01·53-s − 0.398·57-s + 1.71·59-s + 1.01·61-s + 0.620·63-s + 0.225·67-s + 1.04·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1403046404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1403046404\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.89T + 27T^{2} \) |
| 7 | \( 1 + 16.6T + 343T^{2} \) |
| 11 | \( 1 + 19.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 30.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 8.38T + 2.43e4T^{2} \) |
| 31 | \( 1 + 331.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 266.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 320.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 83.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 390.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 779.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 483.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 123.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 187.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 778.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 446.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 94.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 252.T + 9.12e5T^{2} \) |
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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
−9.154119063994364030946419898341, −8.230024549031977261201112575334, −7.32059690637410351753942528953, −6.63514838359359082725961593231, −5.62545057455784191406932348938, −5.21184636563706424518591945284, −3.95132483746709015920113435546, −2.97144898772109666039939184740, −2.00960561430840528515217407686, −0.17202929364149005461743655765,
0.17202929364149005461743655765, 2.00960561430840528515217407686, 2.97144898772109666039939184740, 3.95132483746709015920113435546, 5.21184636563706424518591945284, 5.62545057455784191406932348938, 6.63514838359359082725961593231, 7.32059690637410351753942528953, 8.230024549031977261201112575334, 9.154119063994364030946419898341
