| L(s) = 1 | − 2-s − 1.87·3-s + 4-s + 3.38·5-s + 1.87·6-s − 8-s + 0.513·9-s − 3.38·10-s + 5.76·11-s − 1.87·12-s + 5.06·13-s − 6.35·15-s + 16-s + 1.41·17-s − 0.513·18-s + 5.16·19-s + 3.38·20-s − 5.76·22-s − 3.79·23-s + 1.87·24-s + 6.47·25-s − 5.06·26-s + 4.66·27-s − 7.64·29-s + 6.35·30-s − 7.47·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.08·3-s + 0.5·4-s + 1.51·5-s + 0.765·6-s − 0.353·8-s + 0.171·9-s − 1.07·10-s + 1.73·11-s − 0.541·12-s + 1.40·13-s − 1.63·15-s + 0.250·16-s + 0.343·17-s − 0.121·18-s + 1.18·19-s + 0.757·20-s − 1.22·22-s − 0.790·23-s + 0.382·24-s + 1.29·25-s − 0.993·26-s + 0.896·27-s − 1.42·29-s + 1.15·30-s − 1.34·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.826652390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.826652390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 101 | \( 1 + T \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 3.38T + 5T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 + 3.79T + 23T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 + 7.47T + 31T^{2} \) |
| 37 | \( 1 + 1.40T + 37T^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 9.08T + 47T^{2} \) |
| 53 | \( 1 - 6.05T + 53T^{2} \) |
| 59 | \( 1 - 9.29T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 7.72T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 6.59T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 - 4.64T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.43703774285019076014545172410, −6.90058015666617411984021028543, −6.09378632391895429591002792786, −5.77181689470143402854881497544, −5.48197244720942008379042635045, −4.11831397283239397610636701846, −3.45270107225110093047025018702, −2.20985112106327146213065306181, −1.41481832490267401222041549386, −0.862511143067342268352646750728,
0.862511143067342268352646750728, 1.41481832490267401222041549386, 2.20985112106327146213065306181, 3.45270107225110093047025018702, 4.11831397283239397610636701846, 5.48197244720942008379042635045, 5.77181689470143402854881497544, 6.09378632391895429591002792786, 6.90058015666617411984021028543, 7.43703774285019076014545172410
