| L(s) = 1 | + 2.62·3-s + 5-s + 2.62·7-s + 3.89·9-s + 1.72·13-s + 2.62·15-s − 0.626·17-s + 8.14·19-s + 6.89·21-s − 5.52·23-s + 25-s + 2.35·27-s + 2.89·29-s − 6.89·31-s + 2.62·35-s − 0.896·37-s + 4.54·39-s + 5.25·41-s + 9.52·43-s + 3.89·45-s − 0.270·47-s − 0.103·49-s − 1.64·51-s − 0.896·53-s + 21.4·57-s + 7.04·59-s − 9.40·61-s + ⋯ |
| L(s) = 1 | + 1.51·3-s + 0.447·5-s + 0.992·7-s + 1.29·9-s + 0.479·13-s + 0.678·15-s − 0.151·17-s + 1.86·19-s + 1.50·21-s − 1.15·23-s + 0.200·25-s + 0.453·27-s + 0.537·29-s − 1.23·31-s + 0.443·35-s − 0.147·37-s + 0.727·39-s + 0.820·41-s + 1.45·43-s + 0.580·45-s − 0.0394·47-s − 0.0147·49-s − 0.230·51-s − 0.123·53-s + 2.83·57-s + 0.917·59-s − 1.20·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.593482126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.593482126\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.62T + 3T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 + 0.626T + 17T^{2} \) |
| 19 | \( 1 - 8.14T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 + 6.89T + 31T^{2} \) |
| 37 | \( 1 + 0.896T + 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 - 9.52T + 43T^{2} \) |
| 47 | \( 1 + 0.270T + 47T^{2} \) |
| 53 | \( 1 + 0.896T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 + 9.40T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 9.60T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.186497593559954404664665553944, −7.74176486257966747957368639220, −7.18844278970153240062772196954, −6.02753710279672125882058815533, −5.33004791085783188706903468343, −4.38530430470477934204038453786, −3.63053328764205418069762730188, −2.82349758898379213675780626550, −1.99560729488448757461083600834, −1.23595057029947059887455714779,
1.23595057029947059887455714779, 1.99560729488448757461083600834, 2.82349758898379213675780626550, 3.63053328764205418069762730188, 4.38530430470477934204038453786, 5.33004791085783188706903468343, 6.02753710279672125882058815533, 7.18844278970153240062772196954, 7.74176486257966747957368639220, 8.186497593559954404664665553944
