| L(s) = 1 | + 1.79·3-s − 4.79·7-s + 0.208·9-s − 11-s − 13-s − 3.79·17-s − 2.58·19-s − 8.58·21-s + 0.791·23-s − 5.00·27-s + 2.20·29-s + 0.582·31-s − 1.79·33-s + 6.58·37-s − 1.79·39-s − 10.5·41-s − 10·43-s − 10.5·47-s + 15.9·49-s − 6.79·51-s − 2.37·53-s − 4.62·57-s − 1.41·59-s + 8.79·61-s − 0.999·63-s − 4·67-s + 1.41·69-s + ⋯ |
| L(s) = 1 | + 1.03·3-s − 1.81·7-s + 0.0695·9-s − 0.301·11-s − 0.277·13-s − 0.919·17-s − 0.592·19-s − 1.87·21-s + 0.164·23-s − 0.962·27-s + 0.410·29-s + 0.104·31-s − 0.311·33-s + 1.08·37-s − 0.286·39-s − 1.65·41-s − 1.52·43-s − 1.54·47-s + 2.27·49-s − 0.950·51-s − 0.326·53-s − 0.612·57-s − 0.184·59-s + 1.12·61-s − 0.125·63-s − 0.488·67-s + 0.170·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 0.791T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 - 0.582T + 31T^{2} \) |
| 37 | \( 1 - 6.58T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 + 3.20T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 3.79T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−9.506475633829259115093445921717, −8.648941851317676998996647537364, −7.997520529206816269097327056233, −6.76974666974495771617996427917, −6.38740297205727405645125170880, −5.06403487682614199907566503390, −3.78437682720816143306271942034, −3.06837399513500100572665776072, −2.23557546262456169949715625068, 0,
2.23557546262456169949715625068, 3.06837399513500100572665776072, 3.78437682720816143306271942034, 5.06403487682614199907566503390, 6.38740297205727405645125170880, 6.76974666974495771617996427917, 7.997520529206816269097327056233, 8.648941851317676998996647537364, 9.506475633829259115093445921717
