| L(s) = 1 | + 1.76·3-s − 2.14·5-s − 2.31·7-s + 0.109·9-s + 0.602·11-s + 2.31·13-s − 3.77·15-s − 4.36·17-s + 3.77·19-s − 4.08·21-s − 7.45·23-s − 0.417·25-s − 5.09·27-s + 8.78·29-s − 5.26·31-s + 1.06·33-s + 4.95·35-s − 7.89·37-s + 4.08·39-s − 9.11·41-s + 2.06·43-s − 0.234·45-s − 9.53·47-s − 1.64·49-s − 7.69·51-s − 2.21·53-s − 1.29·55-s + ⋯ |
| L(s) = 1 | + 1.01·3-s − 0.957·5-s − 0.874·7-s + 0.0364·9-s + 0.181·11-s + 0.642·13-s − 0.974·15-s − 1.05·17-s + 0.865·19-s − 0.890·21-s − 1.55·23-s − 0.0835·25-s − 0.980·27-s + 1.63·29-s − 0.946·31-s + 0.185·33-s + 0.837·35-s − 1.29·37-s + 0.653·39-s − 1.42·41-s + 0.315·43-s − 0.0349·45-s − 1.39·47-s − 0.234·49-s − 1.07·51-s − 0.304·53-s − 0.174·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 2.14T + 5T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 - 0.602T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 - 3.77T + 19T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 - 8.78T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 7.89T + 37T^{2} \) |
| 41 | \( 1 + 9.11T + 41T^{2} \) |
| 43 | \( 1 - 2.06T + 43T^{2} \) |
| 47 | \( 1 + 9.53T + 47T^{2} \) |
| 53 | \( 1 + 2.21T + 53T^{2} \) |
| 59 | \( 1 + 5.79T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 7.93T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 83 | \( 1 - 7.18T + 83T^{2} \) |
| 89 | \( 1 - 7.33T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−9.100469513612896547958410808933, −8.361731705135211180012755670983, −7.84238866599879614977430851757, −6.82863946739119819103426721175, −6.08793408877917987868609363509, −4.74030367729740961106552915919, −3.59529958432571237117709391793, −3.31269604395047616232752788618, −1.96882514720437380541949957434, 0,
1.96882514720437380541949957434, 3.31269604395047616232752788618, 3.59529958432571237117709391793, 4.74030367729740961106552915919, 6.08793408877917987868609363509, 6.82863946739119819103426721175, 7.84238866599879614977430851757, 8.361731705135211180012755670983, 9.100469513612896547958410808933
