| L(s) = 1 | + 1.74·3-s + 1.48·5-s − 4.05·7-s + 0.0560·9-s − 5.17·11-s − 4.08·13-s + 2.59·15-s − 4.47·17-s + 4.91·19-s − 7.09·21-s + 9.42·23-s − 2.78·25-s − 5.14·27-s − 8.58·29-s − 3.67·31-s − 9.04·33-s − 6.03·35-s − 7.51·37-s − 7.14·39-s + 3.18·41-s + 3.50·43-s + 0.0833·45-s + 1.35·47-s + 9.46·49-s − 7.81·51-s + 4.98·53-s − 7.68·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 0.664·5-s − 1.53·7-s + 0.0186·9-s − 1.55·11-s − 1.13·13-s + 0.671·15-s − 1.08·17-s + 1.12·19-s − 1.54·21-s + 1.96·23-s − 0.557·25-s − 0.990·27-s − 1.59·29-s − 0.659·31-s − 1.57·33-s − 1.01·35-s − 1.23·37-s − 1.14·39-s + 0.497·41-s + 0.534·43-s + 0.0124·45-s + 0.197·47-s + 1.35·49-s − 1.09·51-s + 0.684·53-s − 1.03·55-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 | \( 1 - 1.74T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 + 4.08T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.91T + 19T^{2} \) |
| 23 | \( 1 - 9.42T + 23T^{2} \) |
| 29 | \( 1 + 8.58T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 - 1.35T + 47T^{2} \) |
| 53 | \( 1 - 4.98T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 4.69T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 + 7.30T + 83T^{2} \) |
| 89 | \( 1 - 8.54T + 89T^{2} \) |
| 97 | \( 1 + 2.52T + 97T^{2} \) |
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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
−9.336262621900150705078341094771, −8.630122077721066588509568638421, −7.37185938332656141861597661056, −7.10953076145862921843666947606, −5.72271177708417913900506566215, −5.18975459108741885731621474103, −3.65710054765632239302303546634, −2.80008509792233966460518414506, −2.28744903388417718466571993622, 0,
2.28744903388417718466571993622, 2.80008509792233966460518414506, 3.65710054765632239302303546634, 5.18975459108741885731621474103, 5.72271177708417913900506566215, 7.10953076145862921843666947606, 7.37185938332656141861597661056, 8.630122077721066588509568638421, 9.336262621900150705078341094771
