| L(s) = 1 | + 4.42·2-s − 3·3-s + 11.5·4-s + 2.84·5-s − 13.2·6-s − 31.6·7-s + 15.8·8-s + 9·9-s + 12.6·10-s − 34.7·12-s − 5.15·13-s − 140.·14-s − 8.54·15-s − 22.6·16-s − 121.·17-s + 39.8·18-s − 34.8·19-s + 32.9·20-s + 95.0·21-s + 116.·23-s − 47.4·24-s − 116.·25-s − 22.7·26-s − 27·27-s − 366.·28-s + 69.4·29-s − 37.8·30-s + ⋯ |
| L(s) = 1 | + 1.56·2-s − 0.577·3-s + 1.44·4-s + 0.254·5-s − 0.903·6-s − 1.71·7-s + 0.699·8-s + 0.333·9-s + 0.398·10-s − 0.835·12-s − 0.109·13-s − 2.67·14-s − 0.147·15-s − 0.353·16-s − 1.73·17-s + 0.521·18-s − 0.420·19-s + 0.368·20-s + 0.988·21-s + 1.05·23-s − 0.403·24-s − 0.935·25-s − 0.171·26-s − 0.192·27-s − 2.47·28-s + 0.444·29-s − 0.230·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4.42T + 8T^{2} \) |
| 5 | \( 1 - 2.84T + 125T^{2} \) |
| 7 | \( 1 + 31.6T + 343T^{2} \) |
| 13 | \( 1 + 5.15T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 420.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 231.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 4.91T + 1.48e5T^{2} \) |
| 59 | \( 1 - 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 556.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 84.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 49.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 383.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 930.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−10.81855941749096344668129816140, −9.832512218723533259092447069774, −8.831444691886818151764838637747, −6.81617544536259023433720957606, −6.59208913294589873558272119908, −5.59724417245814308492542955500, −4.54193665197481135251553781018, −3.51553720050145931323056031409, −2.39876748337441984112988073793, 0,
2.39876748337441984112988073793, 3.51553720050145931323056031409, 4.54193665197481135251553781018, 5.59724417245814308492542955500, 6.59208913294589873558272119908, 6.81617544536259023433720957606, 8.831444691886818151764838637747, 9.832512218723533259092447069774, 10.81855941749096344668129816140
