L(s) = 1 | + 2-s + 2.18·3-s + 4-s + 0.389·5-s + 2.18·6-s + 4.93·7-s + 8-s + 1.77·9-s + 0.389·10-s + 11-s + 2.18·12-s + 0.775·13-s + 4.93·14-s + 0.851·15-s + 16-s − 5.68·17-s + 1.77·18-s − 1.12·19-s + 0.389·20-s + 10.7·21-s + 22-s + 7.98·23-s + 2.18·24-s − 4.84·25-s + 0.775·26-s − 2.66·27-s + 4.93·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s + 0.174·5-s + 0.892·6-s + 1.86·7-s + 0.353·8-s + 0.592·9-s + 0.123·10-s + 0.301·11-s + 0.631·12-s + 0.215·13-s + 1.31·14-s + 0.219·15-s + 0.250·16-s − 1.37·17-s + 0.419·18-s − 0.257·19-s + 0.0870·20-s + 2.35·21-s + 0.213·22-s + 1.66·23-s + 0.446·24-s − 0.969·25-s + 0.152·26-s − 0.513·27-s + 0.932·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.205790351\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.205790351\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 - 0.389T + 5T^{2} \) |
| 7 | \( 1 - 4.93T + 7T^{2} \) |
| 13 | \( 1 - 0.775T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 7.98T + 23T^{2} \) |
| 29 | \( 1 + 9.93T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 - 7.79T + 43T^{2} \) |
| 47 | \( 1 - 2.14T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 + 2.55T + 59T^{2} \) |
| 61 | \( 1 - 0.705T + 61T^{2} \) |
| 67 | \( 1 - 3.66T + 67T^{2} \) |
| 71 | \( 1 + 5.92T + 71T^{2} \) |
| 73 | \( 1 - 6.54T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
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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
−8.299241118446607390274828403612, −7.69147610677617202287793205770, −7.14667631331269232379993504800, −6.04533473134255753353194277985, −5.28902091819003117560181536806, −4.35498491821333225907989816572, −4.03364626256593850052136879689, −2.79203288715337166686481367668, −2.16151152069260841927431680248, −1.39290597667251261470960384384,
1.39290597667251261470960384384, 2.16151152069260841927431680248, 2.79203288715337166686481367668, 4.03364626256593850052136879689, 4.35498491821333225907989816572, 5.28902091819003117560181536806, 6.04533473134255753353194277985, 7.14667631331269232379993504800, 7.69147610677617202287793205770, 8.299241118446607390274828403612