| L(s) = 1 | + 1.67·3-s − 5-s + 4.63·7-s − 0.193·9-s + 4.96·11-s + 5.76·13-s − 1.67·15-s + 6.54·17-s − 4.32·19-s + 7.76·21-s − 2.96·23-s + 25-s − 5.35·27-s − 4.57·29-s + 5.02·31-s + 8.31·33-s − 4.63·35-s − 37-s + 9.66·39-s − 5.35·41-s + 0.574·43-s + 0.193·45-s − 13.5·47-s + 14.5·49-s + 10.9·51-s − 5.53·53-s − 4.96·55-s + ⋯ |
| L(s) = 1 | + 0.967·3-s − 0.447·5-s + 1.75·7-s − 0.0646·9-s + 1.49·11-s + 1.59·13-s − 0.432·15-s + 1.58·17-s − 0.992·19-s + 1.69·21-s − 0.617·23-s + 0.200·25-s − 1.02·27-s − 0.849·29-s + 0.902·31-s + 1.44·33-s − 0.783·35-s − 0.164·37-s + 1.54·39-s − 0.835·41-s + 0.0876·43-s + 0.0289·45-s − 1.98·47-s + 2.07·49-s + 1.53·51-s − 0.760·53-s − 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.432052989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.432052989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 7 | \( 1 - 4.63T + 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 13 | \( 1 - 5.76T + 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 + 2.96T + 23T^{2} \) |
| 29 | \( 1 + 4.57T + 29T^{2} \) |
| 31 | \( 1 - 5.02T + 31T^{2} \) |
| 41 | \( 1 + 5.35T + 41T^{2} \) |
| 43 | \( 1 - 0.574T + 43T^{2} \) |
| 47 | \( 1 + 13.5T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 + 1.54T + 59T^{2} \) |
| 61 | \( 1 - 9.47T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 + 5.98T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 - 2.64T + 97T^{2} \) |
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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
−8.505611056339693907746161944609, −8.199162079917898863725653790231, −7.60642818581709600640377322536, −6.47571199648547292350832518504, −5.72373435547290125203229037126, −4.64936525818490053049030861588, −3.84541987933517038645673354238, −3.33033084219851388236458273043, −1.86606197757820266971392077679, −1.28099535110036857852456719629,
1.28099535110036857852456719629, 1.86606197757820266971392077679, 3.33033084219851388236458273043, 3.84541987933517038645673354238, 4.64936525818490053049030861588, 5.72373435547290125203229037126, 6.47571199648547292350832518504, 7.60642818581709600640377322536, 8.199162079917898863725653790231, 8.505611056339693907746161944609
