| L(s) = 1 | − 2-s + 0.540·3-s + 4-s + 1.92·5-s − 0.540·6-s + 2.85·7-s − 8-s − 2.70·9-s − 1.92·10-s + 6.29·11-s + 0.540·12-s − 5.64·13-s − 2.85·14-s + 1.04·15-s + 16-s + 1.77·17-s + 2.70·18-s + 6.79·19-s + 1.92·20-s + 1.54·21-s − 6.29·22-s − 23-s − 0.540·24-s − 1.28·25-s + 5.64·26-s − 3.08·27-s + 2.85·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.312·3-s + 0.5·4-s + 0.862·5-s − 0.220·6-s + 1.08·7-s − 0.353·8-s − 0.902·9-s − 0.609·10-s + 1.89·11-s + 0.156·12-s − 1.56·13-s − 0.764·14-s + 0.269·15-s + 0.250·16-s + 0.429·17-s + 0.638·18-s + 1.55·19-s + 0.431·20-s + 0.337·21-s − 1.34·22-s − 0.208·23-s − 0.110·24-s − 0.256·25-s + 1.10·26-s − 0.594·27-s + 0.540·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.813844152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.813844152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.540T + 3T^{2} \) |
| 5 | \( 1 - 1.92T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 11 | \( 1 - 6.29T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 - 1.77T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 - 2.88T + 37T^{2} \) |
| 41 | \( 1 - 0.887T + 41T^{2} \) |
| 43 | \( 1 + 5.20T + 43T^{2} \) |
| 47 | \( 1 + 0.232T + 47T^{2} \) |
| 53 | \( 1 + 9.34T + 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 - 0.247T + 61T^{2} \) |
| 67 | \( 1 - 2.98T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 + 8.25T + 73T^{2} \) |
| 79 | \( 1 - 0.989T + 79T^{2} \) |
| 83 | \( 1 + 4.38T + 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−9.763334115549973086110223778239, −8.887422560326546038740093641354, −8.106919157370125137377544588453, −7.36265878622355428400715330340, −6.40188944910318418747487939257, −5.54357507738702509804684787847, −4.62940135890923197611937754291, −3.21307249614394447937254989670, −2.15798894444054046436560488654, −1.18772605783289451063603258964,
1.18772605783289451063603258964, 2.15798894444054046436560488654, 3.21307249614394447937254989670, 4.62940135890923197611937754291, 5.54357507738702509804684787847, 6.40188944910318418747487939257, 7.36265878622355428400715330340, 8.106919157370125137377544588453, 8.887422560326546038740093641354, 9.763334115549973086110223778239
