| L(s) = 1 | + 2-s + 4-s − 3.46·5-s − 1.73·7-s + 8-s − 3.46·10-s − 5.19·13-s − 1.73·14-s + 16-s − 3.46·19-s − 3.46·20-s − 6.92·23-s + 6.99·25-s − 5.19·26-s − 1.73·28-s + 6·29-s − 5·31-s + 32-s + 5.99·35-s − 2·37-s − 3.46·38-s − 3.46·40-s − 6·41-s + 10.3·43-s − 6.92·46-s + 10.3·47-s − 4·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.54·5-s − 0.654·7-s + 0.353·8-s − 1.09·10-s − 1.44·13-s − 0.462·14-s + 0.250·16-s − 0.794·19-s − 0.774·20-s − 1.44·23-s + 1.39·25-s − 1.01·26-s − 0.327·28-s + 1.11·29-s − 0.898·31-s + 0.176·32-s + 1.01·35-s − 0.328·37-s − 0.561·38-s − 0.547·40-s − 0.937·41-s + 1.58·43-s − 1.02·46-s + 1.51·47-s − 0.571·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.114533326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114533326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 + 8.66T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 19T + 97T^{2} \) |
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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
−7.937396916558292396224856616253, −7.14838872334970003639237283629, −6.78139655024936047786398515666, −5.82553276686893459584177611852, −5.00934227530519255257259834233, −4.18980089252404368337048170225, −3.84856019517501652117925220470, −2.90720231452198974664059821956, −2.15040907523938320175611259738, −0.45849701381299715111386127339,
0.45849701381299715111386127339, 2.15040907523938320175611259738, 2.90720231452198974664059821956, 3.84856019517501652117925220470, 4.18980089252404368337048170225, 5.00934227530519255257259834233, 5.82553276686893459584177611852, 6.78139655024936047786398515666, 7.14838872334970003639237283629, 7.937396916558292396224856616253
