| L(s) = 1 | + 4.76·3-s + 15.5·7-s − 4.30·9-s + 74.3·21-s − 207.·23-s − 149.·27-s − 306·29-s + 460.·41-s − 30.9·43-s − 643.·47-s − 99.7·49-s − 40.2·61-s − 67.1·63-s + 1.09e3·67-s − 990.·69-s − 594.·81-s − 1.14e3·83-s − 1.45e3·87-s − 1.38e3·89-s + 378·101-s − 1.98e3·103-s + 1.77e3·107-s + 1.97e3·109-s + ⋯ |
| L(s) = 1 | + 0.916·3-s + 0.842·7-s − 0.159·9-s + 0.772·21-s − 1.88·23-s − 1.06·27-s − 1.95·29-s + 1.75·41-s − 0.109·43-s − 1.99·47-s − 0.290·49-s − 0.0844·61-s − 0.134·63-s + 1.99·67-s − 1.72·69-s − 0.815·81-s − 1.51·83-s − 1.79·87-s − 1.65·89-s + 0.372·101-s − 1.89·103-s + 1.59·107-s + 1.73·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4.76T + 27T^{2} \) |
| 7 | \( 1 - 15.5T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 207.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 306T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 - 460.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 643.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 40.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.09e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−8.464429902389185272703679730418, −8.009817117120681502077901560661, −7.35399996691795161374736349543, −6.12641507865399595098243331562, −5.37638676797058364586026039710, −4.25909504267791003525876855839, −3.51414452409334200763792719101, −2.36633104530424970523466130566, −1.64517511305892543787265414269, 0,
1.64517511305892543787265414269, 2.36633104530424970523466130566, 3.51414452409334200763792719101, 4.25909504267791003525876855839, 5.37638676797058364586026039710, 6.12641507865399595098243331562, 7.35399996691795161374736349543, 8.009817117120681502077901560661, 8.464429902389185272703679730418
