| L(s) = 1 | + 0.414·2-s − 0.335·3-s − 1.82·4-s − 0.139·6-s + 3.31·7-s − 1.58·8-s − 2.88·9-s − 11-s + 0.613·12-s − 6.37·13-s + 1.37·14-s + 2.99·16-s + 0.808·17-s − 1.19·18-s + 19-s − 1.11·21-s − 0.414·22-s + 0.317·23-s + 0.533·24-s − 2.64·26-s + 1.97·27-s − 6.06·28-s + 4.79·29-s − 8.93·31-s + 4.41·32-s + 0.335·33-s + 0.335·34-s + ⋯ |
| L(s) = 1 | + 0.293·2-s − 0.193·3-s − 0.913·4-s − 0.0568·6-s + 1.25·7-s − 0.561·8-s − 0.962·9-s − 0.301·11-s + 0.177·12-s − 1.76·13-s + 0.367·14-s + 0.749·16-s + 0.196·17-s − 0.282·18-s + 0.229·19-s − 0.242·21-s − 0.0884·22-s + 0.0662·23-s + 0.108·24-s − 0.518·26-s + 0.380·27-s − 1.14·28-s + 0.891·29-s − 1.60·31-s + 0.781·32-s + 0.0584·33-s + 0.0575·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.134206255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134206255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 + 0.335T + 3T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 - 0.808T + 17T^{2} \) |
| 23 | \( 1 - 0.317T + 23T^{2} \) |
| 29 | \( 1 - 4.79T + 29T^{2} \) |
| 31 | \( 1 + 8.93T + 31T^{2} \) |
| 37 | \( 1 + 7.03T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 - 0.421T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 7.32T + 67T^{2} \) |
| 71 | \( 1 + 1.69T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 3.53T + 79T^{2} \) |
| 83 | \( 1 - 9.36T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.176353731510166152714665768369, −7.65466003108357747981461339522, −6.81603129148513286326135130080, −5.62703056927444845499990961527, −5.17417137635190008411217319282, −4.82017324604328643199521841576, −3.83723244210393233803827310947, −2.87927375012768380854433592104, −1.97356774737991183811782995119, −0.54477983586144601814835418117,
0.54477983586144601814835418117, 1.97356774737991183811782995119, 2.87927375012768380854433592104, 3.83723244210393233803827310947, 4.82017324604328643199521841576, 5.17417137635190008411217319282, 5.62703056927444845499990961527, 6.81603129148513286326135130080, 7.65466003108357747981461339522, 8.176353731510166152714665768369
