| L(s) = 1 | + 1.07·2-s − 0.838·4-s − 3.04·5-s + 3.44·7-s − 3.05·8-s − 3.28·10-s − 11-s + 2.71·13-s + 3.71·14-s − 1.62·16-s − 3.42·17-s − 1.81·19-s + 2.55·20-s − 1.07·22-s + 6.08·23-s + 4.28·25-s + 2.92·26-s − 2.89·28-s − 0.376·29-s + 4.93·31-s + 4.37·32-s − 3.68·34-s − 10.5·35-s + 3.50·37-s − 1.95·38-s + 9.32·40-s − 5.87·41-s + ⋯ |
| L(s) = 1 | + 0.762·2-s − 0.419·4-s − 1.36·5-s + 1.30·7-s − 1.08·8-s − 1.03·10-s − 0.301·11-s + 0.752·13-s + 0.993·14-s − 0.405·16-s − 0.829·17-s − 0.416·19-s + 0.571·20-s − 0.229·22-s + 1.26·23-s + 0.856·25-s + 0.573·26-s − 0.546·28-s − 0.0699·29-s + 0.887·31-s + 0.772·32-s − 0.632·34-s − 1.77·35-s + 0.575·37-s − 0.317·38-s + 1.47·40-s − 0.917·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.07T + 2T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 + 3.42T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 6.08T + 23T^{2} \) |
| 29 | \( 1 + 0.376T + 29T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 - 3.50T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 - 9.20T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 + 8.15T + 89T^{2} \) |
| 97 | \( 1 - 1.34T + 97T^{2} \) |
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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
−7.990873008301860577797823568786, −6.93258918070019526610559048299, −6.30235011879157869754133257937, −5.06445970146917937113767566159, −4.90877162422926144039123208594, −4.09728937923865556921034505566, −3.54265255354682789859501257173, −2.59412830700236106275415435950, −1.24790399288611501047297934870, 0,
1.24790399288611501047297934870, 2.59412830700236106275415435950, 3.54265255354682789859501257173, 4.09728937923865556921034505566, 4.90877162422926144039123208594, 5.06445970146917937113767566159, 6.30235011879157869754133257937, 6.93258918070019526610559048299, 7.990873008301860577797823568786
