L(s) = 1 | + 0.620·2-s + 0.467·3-s − 7.61·4-s + 5·5-s + 0.290·6-s − 15.6·7-s − 9.68·8-s − 26.7·9-s + 3.10·10-s + 11·11-s − 3.56·12-s + 20.4·13-s − 9.70·14-s + 2.33·15-s + 54.9·16-s − 120.·17-s − 16.6·18-s − 19·19-s − 38.0·20-s − 7.32·21-s + 6.82·22-s − 40.8·23-s − 4.53·24-s + 25·25-s + 12.6·26-s − 25.1·27-s + 119.·28-s + ⋯ |
L(s) = 1 | + 0.219·2-s + 0.0900·3-s − 0.951·4-s + 0.447·5-s + 0.0197·6-s − 0.844·7-s − 0.428·8-s − 0.991·9-s + 0.0980·10-s + 0.301·11-s − 0.0857·12-s + 0.435·13-s − 0.185·14-s + 0.0402·15-s + 0.858·16-s − 1.71·17-s − 0.217·18-s − 0.229·19-s − 0.425·20-s − 0.0760·21-s + 0.0661·22-s − 0.370·23-s − 0.0385·24-s + 0.200·25-s + 0.0955·26-s − 0.179·27-s + 0.804·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.045164708\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.045164708\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
good | 2 | \( 1 - 0.620T + 8T^{2} \) |
| 3 | \( 1 - 0.467T + 27T^{2} \) |
| 7 | \( 1 + 15.6T + 343T^{2} \) |
| 13 | \( 1 - 20.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 40.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 204.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 108.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 143.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 18.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 579.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 83.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 715.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 791.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 498.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 572.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 240.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 700.T + 9.12e5T^{2} \) |
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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
−9.277343797890802548184598471933, −8.927439886006815831965446717525, −8.159567130792477568375351636751, −6.74923366674201065587173358525, −6.09911956241280624191084619837, −5.26045781467760953076037976264, −4.19584361058980836457696395842, −3.34251310391215763920751478071, −2.22734195856135828316217420556, −0.50099925142794268908025464275,
0.50099925142794268908025464275, 2.22734195856135828316217420556, 3.34251310391215763920751478071, 4.19584361058980836457696395842, 5.26045781467760953076037976264, 6.09911956241280624191084619837, 6.74923366674201065587173358525, 8.159567130792477568375351636751, 8.927439886006815831965446717525, 9.277343797890802548184598471933