L(s) = 1 | + 2-s + 3.23·3-s + 4-s − 5-s + 3.23·6-s + 2.78·7-s + 8-s + 7.45·9-s − 10-s + 2.03·11-s + 3.23·12-s + 1.97·13-s + 2.78·14-s − 3.23·15-s + 16-s + 0.661·17-s + 7.45·18-s − 6.72·19-s − 20-s + 9.00·21-s + 2.03·22-s + 5.27·23-s + 3.23·24-s + 25-s + 1.97·26-s + 14.4·27-s + 2.78·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.86·3-s + 0.5·4-s − 0.447·5-s + 1.32·6-s + 1.05·7-s + 0.353·8-s + 2.48·9-s − 0.316·10-s + 0.612·11-s + 0.933·12-s + 0.548·13-s + 0.744·14-s − 0.834·15-s + 0.250·16-s + 0.160·17-s + 1.75·18-s − 1.54·19-s − 0.223·20-s + 1.96·21-s + 0.433·22-s + 1.10·23-s + 0.660·24-s + 0.200·25-s + 0.387·26-s + 2.77·27-s + 0.526·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.742256205\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.742256205\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 - 1.97T + 13T^{2} \) |
| 17 | \( 1 - 0.661T + 17T^{2} \) |
| 19 | \( 1 + 6.72T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 + 8.07T + 29T^{2} \) |
| 31 | \( 1 - 2.95T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 5.85T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 + 1.72T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.61T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 0.879T + 71T^{2} \) |
| 73 | \( 1 + 0.818T + 73T^{2} \) |
| 79 | \( 1 - 9.53T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 2.61T + 97T^{2} \) |
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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
−8.381363246459184835328129225363, −7.895962524521362591477115346516, −7.06558850210980501407616794313, −6.49145738444860659549921459607, −5.10458770394387729335756053627, −4.45279300293143102638935873627, −3.69551011777850507780457500462, −3.21240966199929889950194262461, −2.03204571100217777561992785898, −1.52615953563523262277908568044,
1.52615953563523262277908568044, 2.03204571100217777561992785898, 3.21240966199929889950194262461, 3.69551011777850507780457500462, 4.45279300293143102638935873627, 5.10458770394387729335756053627, 6.49145738444860659549921459607, 7.06558850210980501407616794313, 7.895962524521362591477115346516, 8.381363246459184835328129225363