| L(s) = 1 | + 2·2-s − 4.37·3-s + 4·4-s − 8.75·6-s − 31.2·7-s + 8·8-s − 7.81·9-s − 14.9·11-s − 17.5·12-s + 42.6·13-s − 62.4·14-s + 16·16-s + 29.1·17-s − 15.6·18-s − 51.4·19-s + 136.·21-s − 29.8·22-s − 185.·23-s − 35.0·24-s + 85.3·26-s + 152.·27-s − 124.·28-s − 76.9·29-s − 260.·31-s + 32·32-s + 65.2·33-s + 58.2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.842·3-s + 0.5·4-s − 0.595·6-s − 1.68·7-s + 0.353·8-s − 0.289·9-s − 0.408·11-s − 0.421·12-s + 0.910·13-s − 1.19·14-s + 0.250·16-s + 0.415·17-s − 0.204·18-s − 0.621·19-s + 1.42·21-s − 0.288·22-s − 1.68·23-s − 0.297·24-s + 0.643·26-s + 1.08·27-s − 0.842·28-s − 0.492·29-s − 1.50·31-s + 0.176·32-s + 0.344·33-s + 0.293·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.128619098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128619098\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.37T + 27T^{2} \) |
| 7 | \( 1 + 31.2T + 343T^{2} \) |
| 11 | \( 1 + 14.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 260.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 306.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 504.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 196.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 452.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 64.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 136.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 235.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 539.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 353.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 976.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.45e3T + 9.12e5T^{2} \) |
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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
−9.462438953275588910778938142730, −8.530077880692809779085015539004, −7.39044464726601628226926353897, −6.47301408427563654362085513289, −5.93605222294290485975150069222, −5.40234464172210091208847480718, −3.99179129080450620807935755817, −3.38434835787058132208527061461, −2.20122151578521860664808096036, −0.47763148946698205106766724056,
0.47763148946698205106766724056, 2.20122151578521860664808096036, 3.38434835787058132208527061461, 3.99179129080450620807935755817, 5.40234464172210091208847480718, 5.93605222294290485975150069222, 6.47301408427563654362085513289, 7.39044464726601628226926353897, 8.530077880692809779085015539004, 9.462438953275588910778938142730
