| L(s) = 1 | + 4·7-s − 4·11-s + 4·17-s − 4·23-s + 6·29-s − 4·31-s + 8·37-s + 10·41-s + 4·43-s + 4·47-s + 9·49-s − 12·53-s + 4·59-s + 2·61-s − 4·67-s + 8·73-s − 16·77-s + 12·79-s − 4·83-s + 10·89-s − 8·97-s + 2·101-s − 4·103-s − 12·107-s − 2·109-s + 12·113-s + 16·119-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s + 0.970·17-s − 0.834·23-s + 1.11·29-s − 0.718·31-s + 1.31·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 9/7·49-s − 1.64·53-s + 0.520·59-s + 0.256·61-s − 0.488·67-s + 0.936·73-s − 1.82·77-s + 1.35·79-s − 0.439·83-s + 1.05·89-s − 0.812·97-s + 0.199·101-s − 0.394·103-s − 1.16·107-s − 0.191·109-s + 1.12·113-s + 1.46·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.197947626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197947626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.226876458955487965206451000942, −7.919167398419439034107654528927, −7.38800256081297456965591203909, −6.15421204606922070021803661672, −5.46525873318933504623906073832, −4.80344495575906117477491537233, −4.06442993199655403344546353806, −2.85239179698436020117619375903, −2.03268353241543090446682328600, −0.897179834357593733449721741613,
0.897179834357593733449721741613, 2.03268353241543090446682328600, 2.85239179698436020117619375903, 4.06442993199655403344546353806, 4.80344495575906117477491537233, 5.46525873318933504623906073832, 6.15421204606922070021803661672, 7.38800256081297456965591203909, 7.919167398419439034107654528927, 8.226876458955487965206451000942
