| L(s) = 1 | + 9.89·5-s − 28·11-s − 4.24·13-s − 49.4·17-s + 5.65·19-s + 112·23-s − 27·25-s + 154·29-s + 33.9·31-s − 20·37-s − 168.·41-s − 76·43-s − 435.·47-s − 532·53-s − 277.·55-s − 316.·59-s + 168.·61-s − 41.9·65-s − 372·67-s − 168·71-s + 253.·73-s − 64·79-s + 673.·83-s − 490·85-s − 425.·89-s + 56.0·95-s − 1.05e3·97-s + ⋯ |
| L(s) = 1 | + 0.885·5-s − 0.767·11-s − 0.0905·13-s − 0.706·17-s + 0.0683·19-s + 1.01·23-s − 0.215·25-s + 0.986·29-s + 0.196·31-s − 0.0888·37-s − 0.641·41-s − 0.269·43-s − 1.35·47-s − 1.37·53-s − 0.679·55-s − 0.699·59-s + 0.353·61-s − 0.0801·65-s − 0.678·67-s − 0.280·71-s + 0.405·73-s − 0.0911·79-s + 0.890·83-s − 0.625·85-s − 0.506·89-s + 0.0604·95-s − 1.10·97-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 | \( 1 - 9.89T + 125T^{2} \) |
| 11 | \( 1 + 28T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.24T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.65T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154T + 2.43e4T^{2} \) |
| 31 | \( 1 - 33.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20T + 5.06e4T^{2} \) |
| 41 | \( 1 + 168.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 76T + 7.95e4T^{2} \) |
| 47 | \( 1 + 435.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 532T + 1.48e5T^{2} \) |
| 59 | \( 1 + 316.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 168.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 372T + 3.00e5T^{2} \) |
| 71 | \( 1 + 168T + 3.57e5T^{2} \) |
| 73 | \( 1 - 253.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 64T + 4.93e5T^{2} \) |
| 83 | \( 1 - 673.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 425.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.581724898363950078579844365455, −7.81364272873641657179917134561, −6.81411267351109432168168952000, −6.18649973778846270400096383560, −5.21568730164511853723254611776, −4.62495918674340215583497702438, −3.25998491034174242075141536798, −2.41217866266894225025872926617, −1.41015132983476454798730585489, 0,
1.41015132983476454798730585489, 2.41217866266894225025872926617, 3.25998491034174242075141536798, 4.62495918674340215583497702438, 5.21568730164511853723254611776, 6.18649973778846270400096383560, 6.81411267351109432168168952000, 7.81364272873641657179917134561, 8.581724898363950078579844365455
