| L(s) = 1 | + 4.28·2-s − 2.85·3-s + 10.3·4-s + 6.36·5-s − 12.2·6-s − 29.4·7-s + 10.2·8-s − 18.8·9-s + 27.3·10-s − 61.3·11-s − 29.6·12-s + 18.5·13-s − 126.·14-s − 18.1·15-s − 39.1·16-s − 80.9·18-s + 115.·19-s + 66.1·20-s + 83.8·21-s − 263.·22-s + 7.38·23-s − 29.2·24-s − 84.4·25-s + 79.5·26-s + 130.·27-s − 305.·28-s − 164.·29-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.548·3-s + 1.29·4-s + 0.569·5-s − 0.831·6-s − 1.58·7-s + 0.453·8-s − 0.699·9-s + 0.863·10-s − 1.68·11-s − 0.712·12-s + 0.395·13-s − 2.40·14-s − 0.312·15-s − 0.611·16-s − 1.05·18-s + 1.39·19-s + 0.739·20-s + 0.871·21-s − 2.55·22-s + 0.0669·23-s − 0.248·24-s − 0.675·25-s + 0.600·26-s + 0.932·27-s − 2.06·28-s − 1.05·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 - 4.28T + 8T^{2} \) |
| 3 | \( 1 + 2.85T + 27T^{2} \) |
| 5 | \( 1 - 6.36T + 125T^{2} \) |
| 7 | \( 1 + 29.4T + 343T^{2} \) |
| 11 | \( 1 + 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.38T + 1.21e4T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 127.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 158.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 31.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 157.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 166.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 343.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 112.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 984.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 524.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 852.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 201.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 22.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 502.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 680.T + 9.12e5T^{2} \) |
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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
−11.19321496561140884396605412158, −10.15581888790174070642541458905, −9.241960682136902541950362826713, −7.60186484150607411021742420230, −6.26133384294909824008489140312, −5.81714747595891308493470587968, −4.98689887679114743215200727867, −3.37531954117928163134716772537, −2.66941105442484189568232135199, 0,
2.66941105442484189568232135199, 3.37531954117928163134716772537, 4.98689887679114743215200727867, 5.81714747595891308493470587968, 6.26133384294909824008489140312, 7.60186484150607411021742420230, 9.241960682136902541950362826713, 10.15581888790174070642541458905, 11.19321496561140884396605412158
