| L(s) = 1 | + 12.4·5-s + 51.1·11-s − 37.2·13-s + 22.2·17-s − 54.3·19-s − 176.·23-s + 29.9·25-s − 61.0·29-s − 319.·31-s − 315.·37-s − 206.·41-s + 339.·43-s + 142.·47-s − 310.·53-s + 636.·55-s − 281.·59-s + 543.·61-s − 463.·65-s − 479.·67-s − 1.10e3·71-s − 239.·73-s + 1.16e3·79-s − 2.93·83-s + 276.·85-s − 1.27e3·89-s − 676.·95-s − 79.0·97-s + ⋯ |
| L(s) = 1 | + 1.11·5-s + 1.40·11-s − 0.794·13-s + 0.316·17-s − 0.656·19-s − 1.60·23-s + 0.239·25-s − 0.391·29-s − 1.85·31-s − 1.39·37-s − 0.784·41-s + 1.20·43-s + 0.440·47-s − 0.803·53-s + 1.55·55-s − 0.621·59-s + 1.14·61-s − 0.884·65-s − 0.874·67-s − 1.84·71-s − 0.383·73-s + 1.65·79-s − 0.00387·83-s + 0.352·85-s − 1.52·89-s − 0.730·95-s − 0.0827·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 12.4T + 125T^{2} \) |
| 11 | \( 1 - 51.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 61.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 339.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 142.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 281.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 543.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 479.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 239.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 2.93T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 79.0T + 9.12e5T^{2} \) |
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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
−8.805448595766898552985203162745, −7.66784074350945642071579570999, −6.87097750727987642237460499925, −6.05001126702932798208359201871, −5.48874988266167824815658396576, −4.34649364899786603094737264772, −3.50876373457885020965654768839, −2.13333575779049972860808218319, −1.58798610577861464682137493355, 0,
1.58798610577861464682137493355, 2.13333575779049972860808218319, 3.50876373457885020965654768839, 4.34649364899786603094737264772, 5.48874988266167824815658396576, 6.05001126702932798208359201871, 6.87097750727987642237460499925, 7.66784074350945642071579570999, 8.805448595766898552985203162745
