| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s − 6·11-s + 12-s + 5·13-s − 2·14-s − 15-s + 16-s − 18-s + 19-s − 20-s + 2·21-s + 6·22-s − 3·23-s − 24-s + 25-s − 5·26-s + 27-s + 2·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s + 0.288·12-s + 1.38·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s + 1.27·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.562293867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562293867\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−13.13092832483173, −12.96518294164781, −12.22464226025400, −11.66677992374104, −11.28986545046923, −10.82017017670871, −10.35125009063079, −9.964482611875508, −9.469251158599652, −8.620796158683538, −8.334251043430481, −8.096131049550421, −7.773427306725547, −7.037842367174456, −6.561454596273108, −5.956259397910079, −5.234438947267207, −4.927366701071253, −4.117719593418218, −3.640683610502810, −2.939141062702694, −2.455710156449208, −1.912848101826420, −1.056120746514657, −0.5766263867301210,
0.5766263867301210, 1.056120746514657, 1.912848101826420, 2.455710156449208, 2.939141062702694, 3.640683610502810, 4.117719593418218, 4.927366701071253, 5.234438947267207, 5.956259397910079, 6.561454596273108, 7.037842367174456, 7.773427306725547, 8.096131049550421, 8.334251043430481, 8.620796158683538, 9.469251158599652, 9.964482611875508, 10.35125009063079, 10.82017017670871, 11.28986545046923, 11.66677992374104, 12.22464226025400, 12.96518294164781, 13.13092832483173
