| L(s) = 1 | + 2.11·2-s − 3-s + 2.48·4-s − 3.30·5-s − 2.11·6-s − 7-s + 1.02·8-s + 9-s − 7.00·10-s + 2.38·11-s − 2.48·12-s − 6.43·13-s − 2.11·14-s + 3.30·15-s − 2.79·16-s + 0.259·17-s + 2.11·18-s − 0.660·19-s − 8.21·20-s + 21-s + 5.05·22-s + 0.00650·23-s − 1.02·24-s + 5.95·25-s − 13.6·26-s − 27-s − 2.48·28-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 0.577·3-s + 1.24·4-s − 1.47·5-s − 0.864·6-s − 0.377·7-s + 0.362·8-s + 0.333·9-s − 2.21·10-s + 0.719·11-s − 0.717·12-s − 1.78·13-s − 0.565·14-s + 0.854·15-s − 0.699·16-s + 0.0628·17-s + 0.499·18-s − 0.151·19-s − 1.83·20-s + 0.218·21-s + 1.07·22-s + 0.00135·23-s − 0.209·24-s + 1.19·25-s − 2.67·26-s − 0.192·27-s − 0.469·28-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
| \(L(1)\) |
\(\approx\) |
\(1.939885872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939885872\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.11T + 2T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 6.43T + 13T^{2} \) |
| 17 | \( 1 - 0.259T + 17T^{2} \) |
| 19 | \( 1 + 0.660T + 19T^{2} \) |
| 23 | \( 1 - 0.00650T + 23T^{2} \) |
| 29 | \( 1 - 9.64T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 6.37T + 53T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 - 3.06T + 73T^{2} \) |
| 79 | \( 1 + 0.871T + 79T^{2} \) |
| 83 | \( 1 - 9.01T + 83T^{2} \) |
| 89 | \( 1 + 6.23T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.156329859930546458417511228367, −7.43558172514088594115770248440, −6.76470577445237787242316419985, −6.21781711989503645242586372362, −5.20739111774294338197060300796, −4.47013982827864288082934012740, −4.21523486884902958033769412324, −3.20432946318652307668103107259, −2.48120651642448888980658666101, −0.62980928569332822115509465681,
0.62980928569332822115509465681, 2.48120651642448888980658666101, 3.20432946318652307668103107259, 4.21523486884902958033769412324, 4.47013982827864288082934012740, 5.20739111774294338197060300796, 6.21781711989503645242586372362, 6.76470577445237787242316419985, 7.43558172514088594115770248440, 8.156329859930546458417511228367
