L(s) = 1 | − 2-s + 2.32·3-s + 4-s + 5-s − 2.32·6-s + 1.00·7-s − 8-s + 2.39·9-s − 10-s − 0.100·11-s + 2.32·12-s + 4.75·13-s − 1.00·14-s + 2.32·15-s + 16-s + 7.59·17-s − 2.39·18-s + 6.05·19-s + 20-s + 2.32·21-s + 0.100·22-s − 5.89·23-s − 2.32·24-s + 25-s − 4.75·26-s − 1.41·27-s + 1.00·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.34·3-s + 0.5·4-s + 0.447·5-s − 0.947·6-s + 0.378·7-s − 0.353·8-s + 0.797·9-s − 0.316·10-s − 0.0302·11-s + 0.670·12-s + 1.31·13-s − 0.267·14-s + 0.599·15-s + 0.250·16-s + 1.84·17-s − 0.563·18-s + 1.38·19-s + 0.223·20-s + 0.507·21-s + 0.0214·22-s − 1.22·23-s − 0.473·24-s + 0.200·25-s − 0.932·26-s − 0.271·27-s + 0.189·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.006376949\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.006376949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 2.32T + 3T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 + 0.100T + 11T^{2} \) |
| 13 | \( 1 - 4.75T + 13T^{2} \) |
| 17 | \( 1 - 7.59T + 17T^{2} \) |
| 19 | \( 1 - 6.05T + 19T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 - 12.0T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 8.79T + 59T^{2} \) |
| 61 | \( 1 + 9.57T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 + 1.60T + 73T^{2} \) |
| 79 | \( 1 - 0.972T + 79T^{2} \) |
| 83 | \( 1 - 6.67T + 83T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
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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
−8.314136854836832047330819499956, −7.82251160417501267655593712373, −7.56600079385447167444765229893, −6.13768004622665382696267136568, −5.80854984950684063988254455314, −4.53076007138765228035130835840, −3.35857692878697662971079111241, −3.05485066071692560409201500870, −1.81549217095831219392545195490, −1.16566686397888874461010076703,
1.16566686397888874461010076703, 1.81549217095831219392545195490, 3.05485066071692560409201500870, 3.35857692878697662971079111241, 4.53076007138765228035130835840, 5.80854984950684063988254455314, 6.13768004622665382696267136568, 7.56600079385447167444765229893, 7.82251160417501267655593712373, 8.314136854836832047330819499956