| L(s) = 1 | − 3·3-s + 15.7·5-s + 9·9-s + 58.6·11-s − 20.7·13-s − 47.2·15-s + 42.9·17-s + 137.·19-s + 29.0·23-s + 122.·25-s − 27·27-s + 8.68·29-s − 202.·31-s − 176.·33-s − 15.2·37-s + 62.2·39-s + 117.·41-s − 101.·43-s + 141.·45-s + 588.·47-s − 128.·51-s + 404.·53-s + 923.·55-s − 411.·57-s + 10.8·59-s − 894.·61-s − 326.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.40·5-s + 0.333·9-s + 1.60·11-s − 0.442·13-s − 0.812·15-s + 0.612·17-s + 1.65·19-s + 0.263·23-s + 0.980·25-s − 0.192·27-s + 0.0555·29-s − 1.17·31-s − 0.928·33-s − 0.0678·37-s + 0.255·39-s + 0.446·41-s − 0.361·43-s + 0.469·45-s + 1.82·47-s − 0.353·51-s + 1.04·53-s + 2.26·55-s − 0.956·57-s + 0.0239·59-s − 1.87·61-s − 0.622·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.839136796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.839136796\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 15.7T + 125T^{2} \) |
| 11 | \( 1 - 58.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 42.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 137.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 29.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.68T + 2.43e4T^{2} \) |
| 31 | \( 1 + 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 15.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 588.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 404.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 10.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 894.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 703.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 138.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 894.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 681.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 246.T + 9.12e5T^{2} \) |
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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
−9.432854040697014490404578537524, −8.978795520919080559394544190418, −7.47773322220385998315867455332, −6.83906596495618749021088401769, −5.82425884765802671957342047445, −5.47812351039140080937377942163, −4.29109993477914549749763454074, −3.10909520937931433231640881281, −1.76469951380628009306305264804, −0.979833689671748119224820366819,
0.979833689671748119224820366819, 1.76469951380628009306305264804, 3.10909520937931433231640881281, 4.29109993477914549749763454074, 5.47812351039140080937377942163, 5.82425884765802671957342047445, 6.83906596495618749021088401769, 7.47773322220385998315867455332, 8.978795520919080559394544190418, 9.432854040697014490404578537524
