| L(s) = 1 | + 2-s + 4-s + 0.449·7-s + 8-s − 4.89·11-s + 0.449·13-s + 0.449·14-s + 16-s − 4.89·17-s + 7.44·19-s − 4.89·22-s + 2.44·23-s + 0.449·26-s + 0.449·28-s − 2.44·29-s + 4.44·31-s + 32-s − 4.89·34-s + 11.3·37-s + 7.44·38-s + 9·41-s − 2.55·43-s − 4.89·44-s + 2.44·46-s + 10.8·47-s − 6.79·49-s + 0.449·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.169·7-s + 0.353·8-s − 1.47·11-s + 0.124·13-s + 0.120·14-s + 0.250·16-s − 1.18·17-s + 1.70·19-s − 1.04·22-s + 0.510·23-s + 0.0881·26-s + 0.0849·28-s − 0.454·29-s + 0.799·31-s + 0.176·32-s − 0.840·34-s + 1.86·37-s + 1.20·38-s + 1.40·41-s − 0.388·43-s − 0.738·44-s + 0.361·46-s + 1.58·47-s − 0.971·49-s + 0.0623·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.926072227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.926072227\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.449T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 2.55T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 3.55T + 53T^{2} \) |
| 59 | \( 1 + 5.44T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 0.348T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 5.44T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 8.79T + 97T^{2} \) |
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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
−8.164846414120969232636172130379, −7.69321839287051440230918612008, −6.96537987447346786636464105372, −6.07780891616591672343534853690, −5.34507559853550374355485566300, −4.78262709527187612958157049051, −3.91968481714598138165389359233, −2.85781991654177406468475374924, −2.33671544460138395962316787426, −0.880250510110528883048241058948,
0.880250510110528883048241058948, 2.33671544460138395962316787426, 2.85781991654177406468475374924, 3.91968481714598138165389359233, 4.78262709527187612958157049051, 5.34507559853550374355485566300, 6.07780891616591672343534853690, 6.96537987447346786636464105372, 7.69321839287051440230918612008, 8.164846414120969232636172130379
