| L(s) = 1 | + 5.34·2-s − 3·3-s + 20.6·4-s + 5·5-s − 16.0·6-s − 17.2·7-s + 67.4·8-s + 9·9-s + 26.7·10-s − 61.8·12-s + 8.03·13-s − 92.2·14-s − 15·15-s + 195.·16-s − 20.9·17-s + 48.1·18-s + 150.·19-s + 103.·20-s + 51.7·21-s + 150.·23-s − 202.·24-s + 25·25-s + 42.9·26-s − 27·27-s − 355.·28-s − 215.·29-s − 80.2·30-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.57·4-s + 0.447·5-s − 1.09·6-s − 0.931·7-s + 2.97·8-s + 0.333·9-s + 0.845·10-s − 1.48·12-s + 0.171·13-s − 1.76·14-s − 0.258·15-s + 3.05·16-s − 0.299·17-s + 0.630·18-s + 1.82·19-s + 1.15·20-s + 0.537·21-s + 1.36·23-s − 1.72·24-s + 0.200·25-s + 0.324·26-s − 0.192·27-s − 2.39·28-s − 1.37·29-s − 0.488·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.249237343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.249237343\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 5.34T + 8T^{2} \) |
| 7 | \( 1 + 17.2T + 343T^{2} \) |
| 13 | \( 1 - 8.03T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 215.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 11.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 61.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 337.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 5.27T + 1.03e5T^{2} \) |
| 53 | \( 1 - 747.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 492.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 766.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 807.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 516.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 769.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 108.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 280.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 379.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−9.107647706506929907118635035245, −7.48910919666661823649337252826, −6.98342593883051615622079322266, −6.23937895647582191355515965912, −5.47805624185853025382838741469, −5.07927566413406014728237320668, −3.89011763407546398641749332785, −3.24555745355426819042546724602, −2.29832199180087388270507584165, −1.01312022908241089503859133362,
1.01312022908241089503859133362, 2.29832199180087388270507584165, 3.24555745355426819042546724602, 3.89011763407546398641749332785, 5.07927566413406014728237320668, 5.47805624185853025382838741469, 6.23937895647582191355515965912, 6.98342593883051615622079322266, 7.48910919666661823649337252826, 9.107647706506929907118635035245
