| L(s) = 1 | − 3.06·2-s − 3·3-s + 1.36·4-s + 0.675·5-s + 9.18·6-s − 3.38·7-s + 20.3·8-s + 9·9-s − 2.06·10-s − 8.79·11-s − 4.09·12-s + 59.4·13-s + 10.3·14-s − 2.02·15-s − 73.0·16-s + 31.2·17-s − 27.5·18-s − 77.5·19-s + 0.921·20-s + 10.1·21-s + 26.9·22-s + 73.1·23-s − 60.9·24-s − 124.·25-s − 181.·26-s − 27·27-s − 4.62·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.577·3-s + 0.170·4-s + 0.0603·5-s + 0.624·6-s − 0.182·7-s + 0.897·8-s + 0.333·9-s − 0.0653·10-s − 0.240·11-s − 0.0984·12-s + 1.26·13-s + 0.197·14-s − 0.0348·15-s − 1.14·16-s + 0.445·17-s − 0.360·18-s − 0.936·19-s + 0.0102·20-s + 0.105·21-s + 0.260·22-s + 0.662·23-s − 0.518·24-s − 0.996·25-s − 1.37·26-s − 0.192·27-s − 0.0311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 + 3.06T + 8T^{2} \) |
| 5 | \( 1 - 0.675T + 125T^{2} \) |
| 7 | \( 1 + 3.38T + 343T^{2} \) |
| 11 | \( 1 + 8.79T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 61.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 326.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 453.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 480.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 185.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 807.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 129.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 272.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 610.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 886.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 51.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 182.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.35e3T + 9.12e5T^{2} \) |
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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
−11.29728290770596157949804437137, −10.69925077714069041598034539928, −9.699291551564355372952621454759, −8.763962774311562882736863978968, −7.78661866188301715337005269301, −6.60248344227984568182085980443, −5.35442008171096690081168049259, −3.85620392315917456670127358869, −1.57937202228929353460327924591, 0,
1.57937202228929353460327924591, 3.85620392315917456670127358869, 5.35442008171096690081168049259, 6.60248344227984568182085980443, 7.78661866188301715337005269301, 8.763962774311562882736863978968, 9.699291551564355372952621454759, 10.69925077714069041598034539928, 11.29728290770596157949804437137
