| L(s) = 1 | − 1.81·2-s − 0.305·3-s + 1.29·4-s − 3.67·5-s + 0.553·6-s + 1.69·7-s + 1.27·8-s − 2.90·9-s + 6.67·10-s − 0.395·12-s + 4.57·13-s − 3.07·14-s + 1.12·15-s − 4.91·16-s − 2.69·17-s + 5.27·18-s − 7.23·19-s − 4.77·20-s − 0.517·21-s + 6.32·23-s − 0.388·24-s + 8.50·25-s − 8.31·26-s + 1.80·27-s + 2.20·28-s + 4.58·29-s − 2.03·30-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.176·3-s + 0.648·4-s − 1.64·5-s + 0.226·6-s + 0.640·7-s + 0.450·8-s − 0.968·9-s + 2.11·10-s − 0.114·12-s + 1.26·13-s − 0.822·14-s + 0.289·15-s − 1.22·16-s − 0.654·17-s + 1.24·18-s − 1.65·19-s − 1.06·20-s − 0.112·21-s + 1.31·23-s − 0.0793·24-s + 1.70·25-s − 1.62·26-s + 0.346·27-s + 0.415·28-s + 0.850·29-s − 0.371·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3467926263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3467926263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 3 | \( 1 + 0.305T + 3T^{2} \) |
| 5 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 7.23T + 53T^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 - 3.34T + 61T^{2} \) |
| 67 | \( 1 + 6.45T + 67T^{2} \) |
| 71 | \( 1 + 1.84T + 71T^{2} \) |
| 73 | \( 1 - 5.87T + 73T^{2} \) |
| 79 | \( 1 + 4.07T + 79T^{2} \) |
| 83 | \( 1 + 6.07T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.357960680690610179803965678410, −7.86273390629792506452657226964, −6.95176686923659283953571134267, −6.43837501552492762815012722240, −5.19753773249157902123927777642, −4.42728152461724452116854107687, −3.78493205275327048017552023991, −2.71599716230959217998703346964, −1.47550013080576317967949323195, −0.41292943303936962215782315246,
0.41292943303936962215782315246, 1.47550013080576317967949323195, 2.71599716230959217998703346964, 3.78493205275327048017552023991, 4.42728152461724452116854107687, 5.19753773249157902123927777642, 6.43837501552492762815012722240, 6.95176686923659283953571134267, 7.86273390629792506452657226964, 8.357960680690610179803965678410
