| L(s) = 1 | + 0.272·2-s − 3-s − 1.92·4-s − 2.12·5-s − 0.272·6-s − 7-s − 1.06·8-s + 9-s − 0.578·10-s + 3.95·11-s + 1.92·12-s − 6.83·13-s − 0.272·14-s + 2.12·15-s + 3.56·16-s − 3.01·17-s + 0.272·18-s − 4.39·19-s + 4.09·20-s + 21-s + 1.07·22-s − 0.487·23-s + 1.06·24-s − 0.489·25-s − 1.86·26-s − 27-s + 1.92·28-s + ⋯ |
| L(s) = 1 | + 0.192·2-s − 0.577·3-s − 0.962·4-s − 0.949·5-s − 0.111·6-s − 0.377·7-s − 0.377·8-s + 0.333·9-s − 0.182·10-s + 1.19·11-s + 0.555·12-s − 1.89·13-s − 0.0727·14-s + 0.548·15-s + 0.890·16-s − 0.730·17-s + 0.0641·18-s − 1.00·19-s + 0.914·20-s + 0.218·21-s + 0.229·22-s − 0.101·23-s + 0.218·24-s − 0.0978·25-s − 0.365·26-s − 0.192·27-s + 0.363·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2244708534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2244708534\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 0.272T + 2T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + 6.83T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 19 | \( 1 + 4.39T + 19T^{2} \) |
| 23 | \( 1 + 0.487T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 + 6.39T + 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 8.26T + 43T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 + 9.24T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 + 0.904T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 0.491T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−8.549938180606810827740208032004, −7.67132509961256589788810310108, −6.87998088867773947370494782155, −6.35551372118025672252127770022, −5.18611306625406591991134584133, −4.68043557823361395346706575122, −3.99801649340795810593883344706, −3.29734842150607528000303173872, −1.88180852331934948006089064426, −0.26344754511258268359096444477,
0.26344754511258268359096444477, 1.88180852331934948006089064426, 3.29734842150607528000303173872, 3.99801649340795810593883344706, 4.68043557823361395346706575122, 5.18611306625406591991134584133, 6.35551372118025672252127770022, 6.87998088867773947370494782155, 7.67132509961256589788810310108, 8.549938180606810827740208032004
