| L(s) = 1 | + 11.4·5-s + 11.2·7-s + 25.8·11-s + 13·13-s + 20.3·17-s − 154.·19-s − 180.·23-s + 5.69·25-s + 20.4·29-s − 266.·31-s + 128.·35-s + 115.·37-s − 391.·41-s − 151.·43-s − 467.·47-s − 216.·49-s − 79.9·53-s + 295.·55-s − 873.·59-s − 187.·61-s + 148.·65-s + 609.·67-s + 248.·71-s + 852.·73-s + 291.·77-s + 331.·79-s − 435.·83-s + ⋯ |
| L(s) = 1 | + 1.02·5-s + 0.607·7-s + 0.709·11-s + 0.277·13-s + 0.290·17-s − 1.86·19-s − 1.63·23-s + 0.0455·25-s + 0.130·29-s − 1.54·31-s + 0.621·35-s + 0.515·37-s − 1.49·41-s − 0.536·43-s − 1.45·47-s − 0.630·49-s − 0.207·53-s + 0.725·55-s − 1.92·59-s − 0.392·61-s + 0.283·65-s + 1.11·67-s + 0.414·71-s + 1.36·73-s + 0.431·77-s + 0.471·79-s − 0.575·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 11.4T + 125T^{2} \) |
| 7 | \( 1 - 11.2T + 343T^{2} \) |
| 11 | \( 1 - 25.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 467.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 79.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 873.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 187.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 609.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 248.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 852.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 331.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 435.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 259.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−8.443700707090205612627363439090, −7.891730555165800029015461791361, −6.59817852038634601254375141455, −6.21034450939501809755487954757, −5.30153719261347299341503075442, −4.37454590335622611558369580232, −3.48539040820367648539553273821, −1.99257318056707754590788133080, −1.67773373538395540818303012771, 0,
1.67773373538395540818303012771, 1.99257318056707754590788133080, 3.48539040820367648539553273821, 4.37454590335622611558369580232, 5.30153719261347299341503075442, 6.21034450939501809755487954757, 6.59817852038634601254375141455, 7.891730555165800029015461791361, 8.443700707090205612627363439090
