| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s + 3·11-s + 7·17-s − 19-s − 6·21-s + 2·23-s − 9·27-s + 8·29-s − 10·31-s − 9·33-s + 2·37-s − 3·41-s + 4·43-s − 12·47-s − 3·49-s − 21·51-s + 2·53-s + 3·57-s − 61-s + 12·63-s + 11·67-s − 6·69-s + 8·71-s + 5·73-s + 6·77-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s + 0.904·11-s + 1.69·17-s − 0.229·19-s − 1.30·21-s + 0.417·23-s − 1.73·27-s + 1.48·29-s − 1.79·31-s − 1.56·33-s + 0.328·37-s − 0.468·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s − 2.94·51-s + 0.274·53-s + 0.397·57-s − 0.128·61-s + 1.51·63-s + 1.34·67-s − 0.722·69-s + 0.949·71-s + 0.585·73-s + 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.617702712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617702712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.52095794031614, −14.79881495100919, −14.39565137052921, −13.92201659729836, −12.87337600052801, −12.64104563509937, −11.99347288959985, −11.63066510381783, −11.13516709058480, −10.68020710135019, −10.02123135700380, −9.594967379118316, −8.800247134081762, −8.011036925235013, −7.525224856769735, −6.718948629148821, −6.407926309182654, −5.657576868383428, −5.129165102124537, −4.758707925254953, −3.909071824854000, −3.285947125906266, −1.958977905970764, −1.251972299332568, −0.6531436790771953,
0.6531436790771953, 1.251972299332568, 1.958977905970764, 3.285947125906266, 3.909071824854000, 4.758707925254953, 5.129165102124537, 5.657576868383428, 6.407926309182654, 6.718948629148821, 7.525224856769735, 8.011036925235013, 8.800247134081762, 9.594967379118316, 10.02123135700380, 10.68020710135019, 11.13516709058480, 11.63066510381783, 11.99347288959985, 12.64104563509937, 12.87337600052801, 13.92201659729836, 14.39565137052921, 14.79881495100919, 15.52095794031614
