| L(s) = 1 | + 0.369·2-s − 7.86·4-s + 5·5-s − 5.85·8-s + 1.84·10-s − 30.6·11-s − 36.4·13-s + 60.7·16-s − 79.7·17-s + 152.·19-s − 39.3·20-s − 11.3·22-s − 22.2·23-s + 25·25-s − 13.4·26-s − 101.·29-s − 249.·31-s + 69.3·32-s − 29.4·34-s + 7.55·37-s + 56.2·38-s − 29.2·40-s + 142.·41-s − 237.·43-s + 240.·44-s − 8.20·46-s − 331.·47-s + ⋯ |
| L(s) = 1 | + 0.130·2-s − 0.982·4-s + 0.447·5-s − 0.258·8-s + 0.0583·10-s − 0.839·11-s − 0.777·13-s + 0.949·16-s − 1.13·17-s + 1.84·19-s − 0.439·20-s − 0.109·22-s − 0.201·23-s + 0.200·25-s − 0.101·26-s − 0.648·29-s − 1.44·31-s + 0.382·32-s − 0.148·34-s + 0.0335·37-s + 0.240·38-s − 0.115·40-s + 0.541·41-s − 0.842·43-s + 0.825·44-s − 0.0263·46-s − 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.118847369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118847369\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.369T + 8T^{2} \) |
| 11 | \( 1 + 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 22.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 101.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 249.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 7.55T + 5.06e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 237.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 331.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 487.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 717.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 354.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 57.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 696.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 261.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 271.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 681.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 160.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 167.T + 9.12e5T^{2} \) |
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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
−8.901595118583710027067535989523, −7.85443593647307058239189967144, −7.33726220635100367663697837872, −6.19988242743820159586912364852, −5.22564648764735014168382794563, −4.98320618262736929124590564255, −3.82334162062850331270272523718, −2.92284695828335124406143698803, −1.82684102205466336176243864705, −0.46085740632888487602078264595,
0.46085740632888487602078264595, 1.82684102205466336176243864705, 2.92284695828335124406143698803, 3.82334162062850331270272523718, 4.98320618262736929124590564255, 5.22564648764735014168382794563, 6.19988242743820159586912364852, 7.33726220635100367663697837872, 7.85443593647307058239189967144, 8.901595118583710027067535989523
