| L(s) = 1 | + 2-s − 3-s + 4-s − 1.12·5-s − 6-s − 0.350·7-s + 8-s + 9-s − 1.12·10-s − 2.98·11-s − 12-s + 2.39·13-s − 0.350·14-s + 1.12·15-s + 16-s − 4.11·17-s + 18-s + 8.36·19-s − 1.12·20-s + 0.350·21-s − 2.98·22-s − 23-s − 24-s − 3.73·25-s + 2.39·26-s − 27-s − 0.350·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.503·5-s − 0.408·6-s − 0.132·7-s + 0.353·8-s + 0.333·9-s − 0.356·10-s − 0.901·11-s − 0.288·12-s + 0.664·13-s − 0.0936·14-s + 0.290·15-s + 0.250·16-s − 0.998·17-s + 0.235·18-s + 1.91·19-s − 0.251·20-s + 0.0764·21-s − 0.637·22-s − 0.208·23-s − 0.204·24-s − 0.746·25-s + 0.469·26-s − 0.192·27-s − 0.0662·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.011787513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011787513\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 0.350T + 7T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 - 8.36T + 19T^{2} \) |
| 31 | \( 1 - 7.08T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 - 8.02T + 41T^{2} \) |
| 43 | \( 1 - 5.37T + 43T^{2} \) |
| 47 | \( 1 + 1.42T + 47T^{2} \) |
| 53 | \( 1 - 1.66T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 4.56T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 + 7.49T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.139082452161990939872671744612, −7.68031456720024839113893264573, −6.84537093591514003515926713445, −6.14894085166250891529018400462, −5.37896614293802357743846322803, −4.78958923365078245463753222779, −3.88789230079512920320002557060, −3.16492370049393667878960154546, −2.10351093880690011808811823647, −0.74554119185062501919586894212,
0.74554119185062501919586894212, 2.10351093880690011808811823647, 3.16492370049393667878960154546, 3.88789230079512920320002557060, 4.78958923365078245463753222779, 5.37896614293802357743846322803, 6.14894085166250891529018400462, 6.84537093591514003515926713445, 7.68031456720024839113893264573, 8.139082452161990939872671744612
