| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 3·11-s + 13-s − 14-s + 16-s + 2·19-s − 3·22-s + 3·23-s − 5·25-s − 26-s + 28-s + 5·31-s − 32-s − 7·37-s − 2·38-s − 3·41-s + 8·43-s + 3·44-s − 3·46-s + 3·47-s + 49-s + 5·50-s + 52-s + 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.639·22-s + 0.625·23-s − 25-s − 0.196·26-s + 0.188·28-s + 0.898·31-s − 0.176·32-s − 1.15·37-s − 0.324·38-s − 0.468·41-s + 1.21·43-s + 0.452·44-s − 0.442·46-s + 0.437·47-s + 1/7·49-s + 0.707·50-s + 0.138·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387653705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387653705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.220725747608034123664354362313, −8.741104533844239670652837381474, −7.83867260492541076636187152090, −7.11996296423595540415557652746, −6.29215796679279225700527276455, −5.42538218373963254725865788850, −4.29341565444102448956917987575, −3.31582010907805915748552075894, −2.05606432181352469404790479184, −0.957671658928271127099414366745,
0.957671658928271127099414366745, 2.05606432181352469404790479184, 3.31582010907805915748552075894, 4.29341565444102448956917987575, 5.42538218373963254725865788850, 6.29215796679279225700527276455, 7.11996296423595540415557652746, 7.83867260492541076636187152090, 8.741104533844239670652837381474, 9.220725747608034123664354362313
