L(s) = 1 | − 2-s − 3.27·3-s + 4-s + 0.401·5-s + 3.27·6-s − 4.68·7-s − 8-s + 7.75·9-s − 0.401·10-s − 1.05·11-s − 3.27·12-s − 6.56·13-s + 4.68·14-s − 1.31·15-s + 16-s + 5.05·17-s − 7.75·18-s + 4.70·19-s + 0.401·20-s + 15.3·21-s + 1.05·22-s + 5.86·23-s + 3.27·24-s − 4.83·25-s + 6.56·26-s − 15.5·27-s − 4.68·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.89·3-s + 0.5·4-s + 0.179·5-s + 1.33·6-s − 1.76·7-s − 0.353·8-s + 2.58·9-s − 0.127·10-s − 0.318·11-s − 0.946·12-s − 1.82·13-s + 1.25·14-s − 0.340·15-s + 0.250·16-s + 1.22·17-s − 1.82·18-s + 1.07·19-s + 0.0898·20-s + 3.34·21-s + 0.225·22-s + 1.22·23-s + 0.669·24-s − 0.967·25-s + 1.28·26-s − 2.99·27-s − 0.884·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1809830572\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1809830572\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 - 0.401T + 5T^{2} \) |
| 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 + 6.56T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 - 4.70T + 19T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 + 2.64T + 29T^{2} \) |
| 37 | \( 1 - 2.11T + 37T^{2} \) |
| 41 | \( 1 + 4.13T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 6.65T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 5.66T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 1.67T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 0.247T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
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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
−7.65744738953923804080735181228, −7.23662796196966308580046889911, −6.73274426021122764911044736540, −5.94351711676647547681624270325, −5.43375954240564950918919968960, −4.81637394446579047542000014396, −3.55376724412726394300667019316, −2.74437777309405584092463653566, −1.37381019414980430981726051054, −0.29024589092425653621035243721,
0.29024589092425653621035243721, 1.37381019414980430981726051054, 2.74437777309405584092463653566, 3.55376724412726394300667019316, 4.81637394446579047542000014396, 5.43375954240564950918919968960, 5.94351711676647547681624270325, 6.73274426021122764911044736540, 7.23662796196966308580046889911, 7.65744738953923804080735181228