| L(s) = 1 | + 3·3-s − 13.9·7-s + 9·9-s − 43.3·11-s − 25.3·13-s − 85.3·17-s − 92.3·19-s − 41.9·21-s − 31.3·23-s + 27·27-s − 220.·29-s + 111.·31-s − 130.·33-s − 123.·37-s − 76.1·39-s + 431.·41-s + 278.·43-s + 340.·47-s − 147.·49-s − 256.·51-s + 367.·53-s − 277.·57-s + 638.·59-s − 195.·61-s − 125.·63-s − 11.5·67-s − 94.1·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 0.333·9-s − 1.18·11-s − 0.541·13-s − 1.21·17-s − 1.11·19-s − 0.436·21-s − 0.284·23-s + 0.192·27-s − 1.40·29-s + 0.648·31-s − 0.686·33-s − 0.550·37-s − 0.312·39-s + 1.64·41-s + 0.987·43-s + 1.05·47-s − 0.428·49-s − 0.703·51-s + 0.953·53-s − 0.643·57-s + 1.40·59-s − 0.410·61-s − 0.251·63-s − 0.0210·67-s − 0.164·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.280887739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280887739\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 13.9T + 343T^{2} \) |
| 11 | \( 1 + 43.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 25.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 85.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 31.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 111.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 123.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 431.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 340.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 367.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 638.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 195.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 11.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 110.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 734.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 195.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 317.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 993.T + 9.12e5T^{2} \) |
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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
−8.692625058385572126592051220598, −7.82259016247202405591676824111, −7.18710031596313882700955787337, −6.35245474683657536491433367613, −5.50951538281375425805131883491, −4.48822481878406901004897580615, −3.76038522774618983886155253810, −2.57221589510109614415195856536, −2.19502395180176969973477176158, −0.45243026966405474420383736462,
0.45243026966405474420383736462, 2.19502395180176969973477176158, 2.57221589510109614415195856536, 3.76038522774618983886155253810, 4.48822481878406901004897580615, 5.50951538281375425805131883491, 6.35245474683657536491433367613, 7.18710031596313882700955787337, 7.82259016247202405591676824111, 8.692625058385572126592051220598
