| L(s) = 1 | − 2·2-s + 1.05·3-s + 4·4-s + 9.53·5-s − 2.11·6-s + 7·7-s − 8·8-s − 25.8·9-s − 19.0·10-s − 42.2·11-s + 4.22·12-s − 14·14-s + 10.0·15-s + 16·16-s − 12.2·17-s + 51.7·18-s + 64.7·19-s + 38.1·20-s + 7.39·21-s + 84.4·22-s + 164.·23-s − 8.44·24-s − 34.0·25-s − 55.8·27-s + 28·28-s + 172.·29-s − 20.1·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.203·3-s + 0.5·4-s + 0.852·5-s − 0.143·6-s + 0.377·7-s − 0.353·8-s − 0.958·9-s − 0.602·10-s − 1.15·11-s + 0.101·12-s − 0.267·14-s + 0.173·15-s + 0.250·16-s − 0.174·17-s + 0.677·18-s + 0.781·19-s + 0.426·20-s + 0.0768·21-s + 0.818·22-s + 1.49·23-s − 0.0718·24-s − 0.272·25-s − 0.398·27-s + 0.188·28-s + 1.10·29-s − 0.122·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.616989497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.616989497\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.05T + 27T^{2} \) |
| 5 | \( 1 - 9.53T + 125T^{2} \) |
| 11 | \( 1 + 42.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 12.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 172.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 421.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 271.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 225.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 202.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 583.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 389.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 108.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 858.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 107.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 736.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 831.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.54e3T + 9.12e5T^{2} \) |
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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
−8.667271487898916385782102132721, −8.000560667548145352366876992828, −7.25337162845465196881544589499, −6.35906251686873871506171333132, −5.41440514640839313272585829811, −5.04627424693228619913655841821, −3.38417781484022799872608362105, −2.62429480854996072247265440810, −1.81866511433383652106493663920, −0.60859713202084822677261850208,
0.60859713202084822677261850208, 1.81866511433383652106493663920, 2.62429480854996072247265440810, 3.38417781484022799872608362105, 5.04627424693228619913655841821, 5.41440514640839313272585829811, 6.35906251686873871506171333132, 7.25337162845465196881544589499, 8.000560667548145352366876992828, 8.667271487898916385782102132721
