| L(s) = 1 | + 2.79·2-s + 10.1·3-s − 0.187·4-s + 5.45·5-s + 28.3·6-s + 7.73·7-s − 22.8·8-s + 75.9·9-s + 15.2·10-s − 11·11-s − 1.90·12-s + 21.6·14-s + 55.3·15-s − 62.4·16-s − 18.5·17-s + 212.·18-s + 65.8·19-s − 1.02·20-s + 78.4·21-s − 30.7·22-s + 38.8·23-s − 232.·24-s − 95.2·25-s + 496.·27-s − 1.45·28-s + 16.1·29-s + 154.·30-s + ⋯ |
| L(s) = 1 | + 0.988·2-s + 1.95·3-s − 0.0234·4-s + 0.488·5-s + 1.92·6-s + 0.417·7-s − 1.01·8-s + 2.81·9-s + 0.482·10-s − 0.301·11-s − 0.0458·12-s + 0.412·14-s + 0.953·15-s − 0.975·16-s − 0.264·17-s + 2.77·18-s + 0.794·19-s − 0.0114·20-s + 0.815·21-s − 0.297·22-s + 0.352·23-s − 1.97·24-s − 0.761·25-s + 3.53·27-s − 0.00980·28-s + 0.103·29-s + 0.941·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.597314207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.597314207\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.79T + 8T^{2} \) |
| 3 | \( 1 - 10.1T + 27T^{2} \) |
| 5 | \( 1 - 5.45T + 125T^{2} \) |
| 7 | \( 1 - 7.73T + 343T^{2} \) |
| 17 | \( 1 + 18.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 16.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 100.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 460.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 93.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 801.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 344.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 186.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 820.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 502.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 199.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 535.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 462.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.010874451676351341252921540198, −8.034577316897183668340894944789, −7.56287798947757692754651361753, −6.45427563313690936908379812192, −5.45597643699313937623158127902, −4.44445876108630620976383225419, −3.95405514109144655594252653171, −2.85150520020241140738353438590, −2.44824904339642465972309649467, −1.17095680702358507897293878035,
1.17095680702358507897293878035, 2.44824904339642465972309649467, 2.85150520020241140738353438590, 3.95405514109144655594252653171, 4.44445876108630620976383225419, 5.45597643699313937623158127902, 6.45427563313690936908379812192, 7.56287798947757692754651361753, 8.034577316897183668340894944789, 9.010874451676351341252921540198
