| L(s) = 1 | + 2-s + 1.29·3-s + 4-s + 5-s + 1.29·6-s + 3.00·7-s + 8-s − 1.33·9-s + 10-s − 2.29·11-s + 1.29·12-s − 1.42·13-s + 3.00·14-s + 1.29·15-s + 16-s + 4.62·17-s − 1.33·18-s − 1.26·19-s + 20-s + 3.87·21-s − 2.29·22-s + 0.959·23-s + 1.29·24-s + 25-s − 1.42·26-s − 5.59·27-s + 3.00·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.744·3-s + 0.5·4-s + 0.447·5-s + 0.526·6-s + 1.13·7-s + 0.353·8-s − 0.445·9-s + 0.316·10-s − 0.692·11-s + 0.372·12-s − 0.395·13-s + 0.802·14-s + 0.333·15-s + 0.250·16-s + 1.12·17-s − 0.314·18-s − 0.291·19-s + 0.223·20-s + 0.845·21-s − 0.489·22-s + 0.200·23-s + 0.263·24-s + 0.200·25-s − 0.279·26-s − 1.07·27-s + 0.567·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.351487329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.351487329\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.29T + 3T^{2} \) |
| 7 | \( 1 - 3.00T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 0.959T + 23T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 + 0.178T + 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 - 6.01T + 47T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 + 4.05T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 - 1.59T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−7.74327099658873021569588765628, −7.29833929541314074020602346172, −6.25761388134413178105088345985, −5.46466692177250598368969478083, −5.15837232016109630432699778712, −4.25597415135228405274241627826, −3.45464148413428629895887303761, −2.54738555746432363253756187623, −2.18224730444433929184577406680, −1.00951386044746091984105672294,
1.00951386044746091984105672294, 2.18224730444433929184577406680, 2.54738555746432363253756187623, 3.45464148413428629895887303761, 4.25597415135228405274241627826, 5.15837232016109630432699778712, 5.46466692177250598368969478083, 6.25761388134413178105088345985, 7.29833929541314074020602346172, 7.74327099658873021569588765628
