| L(s) = 1 | + 3-s − 7-s + 9-s + 4·11-s − 6·13-s + 2·17-s − 6·19-s − 21-s − 2·23-s + 27-s + 6·29-s + 2·31-s + 4·33-s − 4·37-s − 6·39-s + 8·41-s + 4·43-s − 4·47-s + 49-s + 2·51-s + 6·53-s − 6·57-s − 4·59-s + 14·61-s − 63-s − 4·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.485·17-s − 1.37·19-s − 0.218·21-s − 0.417·23-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.696·33-s − 0.657·37-s − 0.960·39-s + 1.24·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.794·57-s − 0.520·59-s + 1.79·61-s − 0.125·63-s − 0.488·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.265738668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.265738668\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.77264127330373631693274024283, −7.10627276092203245515839528081, −6.53484240872710294433637069234, −5.85310297812731078570638164177, −4.78437690958523935970356548425, −4.26961424527227245356622570043, −3.48216727550901432788578145099, −2.59326590261832668004147037046, −1.95424826830076372378178266015, −0.70400628411372527229278405618,
0.70400628411372527229278405618, 1.95424826830076372378178266015, 2.59326590261832668004147037046, 3.48216727550901432788578145099, 4.26961424527227245356622570043, 4.78437690958523935970356548425, 5.85310297812731078570638164177, 6.53484240872710294433637069234, 7.10627276092203245515839528081, 7.77264127330373631693274024283
