L(s) = 1 | − 3-s − 5-s − 3.33·7-s + 9-s − 6.35·11-s + 3.86·13-s + 15-s + 5.25·17-s − 2.53·19-s + 3.33·21-s + 7.99·23-s + 25-s − 27-s − 7.12·29-s − 31-s + 6.35·33-s + 3.33·35-s + 2.79·37-s − 3.86·39-s + 4.20·41-s − 4.73·43-s − 45-s − 1.63·47-s + 4.10·49-s − 5.25·51-s − 4.37·53-s + 6.35·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.25·7-s + 0.333·9-s − 1.91·11-s + 1.07·13-s + 0.258·15-s + 1.27·17-s − 0.581·19-s + 0.727·21-s + 1.66·23-s + 0.200·25-s − 0.192·27-s − 1.32·29-s − 0.179·31-s + 1.10·33-s + 0.563·35-s + 0.459·37-s − 0.619·39-s + 0.656·41-s − 0.722·43-s − 0.149·45-s − 0.238·47-s + 0.585·49-s − 0.736·51-s − 0.600·53-s + 0.857·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 + 6.35T + 11T^{2} \) |
| 13 | \( 1 - 3.86T + 13T^{2} \) |
| 17 | \( 1 - 5.25T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 23 | \( 1 - 7.99T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 37 | \( 1 - 2.79T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 + 1.63T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 - 0.231T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 9.42T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
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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
−7.66471361358399754820759794105, −6.74518790017654987087312436693, −6.18824157611908336310075169020, −5.38325599034459390381763972360, −4.96958340997735078920689435625, −3.70677294095359529843873470308, −3.30624608501298123819685132627, −2.39505879620166464064007809387, −0.966124232233696994084201142579, 0,
0.966124232233696994084201142579, 2.39505879620166464064007809387, 3.30624608501298123819685132627, 3.70677294095359529843873470308, 4.96958340997735078920689435625, 5.38325599034459390381763972360, 6.18824157611908336310075169020, 6.74518790017654987087312436693, 7.66471361358399754820759794105