| L(s) = 1 | − 3-s + 9-s + 11-s − 4·13-s − 2·17-s − 2·19-s + 8·23-s − 27-s − 4·29-s + 4·31-s − 33-s + 2·37-s + 4·39-s + 12·41-s − 8·43-s − 4·47-s + 2·51-s + 10·53-s + 2·57-s − 4·59-s − 6·61-s − 12·67-s − 8·69-s − 16·73-s + 6·79-s + 81-s − 2·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.458·19-s + 1.66·23-s − 0.192·27-s − 0.742·29-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.640·39-s + 1.87·41-s − 1.21·43-s − 0.583·47-s + 0.280·51-s + 1.37·53-s + 0.264·57-s − 0.520·59-s − 0.768·61-s − 1.46·67-s − 0.963·69-s − 1.87·73-s + 0.675·79-s + 1/9·81-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092684025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092684025\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.08633042057635, −12.84660642138358, −12.33502070854715, −11.72387468777261, −11.48115798006487, −10.89472907415846, −10.48896858843826, −9.969832210391624, −9.432558475197417, −9.012991238416516, −8.566565779230533, −7.791370270181572, −7.367790606114577, −6.934472321158397, −6.445423701037616, −5.865261077628328, −5.375948730459344, −4.704915423902741, −4.468719948939913, −3.825029656057960, −2.920074375082567, −2.642435884922989, −1.771342507372445, −1.164132929109741, −0.3352155247868427,
0.3352155247868427, 1.164132929109741, 1.771342507372445, 2.642435884922989, 2.920074375082567, 3.825029656057960, 4.468719948939913, 4.704915423902741, 5.375948730459344, 5.865261077628328, 6.445423701037616, 6.934472321158397, 7.367790606114577, 7.791370270181572, 8.566565779230533, 9.012991238416516, 9.432558475197417, 9.969832210391624, 10.48896858843826, 10.89472907415846, 11.48115798006487, 11.72387468777261, 12.33502070854715, 12.84660642138358, 13.08633042057635
