| L(s) = 1 | − 0.323·3-s − 5-s − 2.68·7-s − 2.89·9-s − 4.66·13-s + 0.323·15-s − 4.62·17-s + 4.34·19-s + 0.867·21-s − 2.77·23-s + 25-s + 1.90·27-s − 3.01·29-s − 2.38·31-s + 2.68·35-s − 10.6·37-s + 1.50·39-s − 2.21·41-s + 7.06·43-s + 2.89·45-s − 4.36·47-s + 0.210·49-s + 1.49·51-s − 6.33·53-s − 1.40·57-s − 11.7·59-s − 3.98·61-s + ⋯ |
| L(s) = 1 | − 0.186·3-s − 0.447·5-s − 1.01·7-s − 0.965·9-s − 1.29·13-s + 0.0834·15-s − 1.12·17-s + 0.995·19-s + 0.189·21-s − 0.578·23-s + 0.200·25-s + 0.366·27-s − 0.559·29-s − 0.428·31-s + 0.453·35-s − 1.74·37-s + 0.241·39-s − 0.345·41-s + 1.07·43-s + 0.431·45-s − 0.636·47-s + 0.0300·49-s + 0.209·51-s − 0.870·53-s − 0.185·57-s − 1.52·59-s − 0.509·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1298038665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1298038665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.323T + 3T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 3.01T + 29T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 2.21T + 41T^{2} \) |
| 43 | \( 1 - 7.06T + 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 + 6.33T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 - 1.02T + 73T^{2} \) |
| 79 | \( 1 + 3.50T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.59432267799500207184932896912, −6.99481009494947048185941768464, −6.37523981963957731876496073407, −5.62064184343880994060648505212, −4.99432621925694533229005367638, −4.19500899658707551557245390840, −3.25882478556172795917413210906, −2.81986066440711716164405209266, −1.79234879653079756343672893153, −0.16185084909528918749016138006,
0.16185084909528918749016138006, 1.79234879653079756343672893153, 2.81986066440711716164405209266, 3.25882478556172795917413210906, 4.19500899658707551557245390840, 4.99432621925694533229005367638, 5.62064184343880994060648505212, 6.37523981963957731876496073407, 6.99481009494947048185941768464, 7.59432267799500207184932896912
