L(s) = 1 | + 1.15·2-s + 3-s − 0.656·4-s − 3.68·5-s + 1.15·6-s − 7-s − 3.07·8-s + 9-s − 4.27·10-s + 2.54·11-s − 0.656·12-s − 1.34·13-s − 1.15·14-s − 3.68·15-s − 2.25·16-s − 5.62·17-s + 1.15·18-s − 4.34·19-s + 2.41·20-s − 21-s + 2.95·22-s + 4.94·23-s − 3.07·24-s + 8.57·25-s − 1.55·26-s + 27-s + 0.656·28-s + ⋯ |
L(s) = 1 | + 0.819·2-s + 0.577·3-s − 0.328·4-s − 1.64·5-s + 0.473·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s − 1.35·10-s + 0.767·11-s − 0.189·12-s − 0.371·13-s − 0.309·14-s − 0.951·15-s − 0.564·16-s − 1.36·17-s + 0.273·18-s − 0.997·19-s + 0.540·20-s − 0.218·21-s + 0.628·22-s + 1.03·23-s − 0.628·24-s + 1.71·25-s − 0.304·26-s + 0.192·27-s + 0.124·28-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
\(L(1)\) |
\(\approx\) |
\(1.450145500\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.450145500\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.15T + 2T^{2} \) |
| 5 | \( 1 + 3.68T + 5T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 - 4.94T + 23T^{2} \) |
| 29 | \( 1 - 2.64T + 29T^{2} \) |
| 31 | \( 1 + 4.32T + 31T^{2} \) |
| 37 | \( 1 - 6.01T + 37T^{2} \) |
| 41 | \( 1 + 3.45T + 41T^{2} \) |
| 43 | \( 1 - 2.63T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 - 9.07T + 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 - 1.93T + 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 0.0108T + 79T^{2} \) |
| 83 | \( 1 + 4.86T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 6.34T + 97T^{2} \) |
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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
−8.563544304455625275261764919554, −7.74544652508325868715628394179, −6.84673778174044601773381981508, −6.43905895974355756267404135720, −5.11246027075057276079437217649, −4.39270631729122541929696421422, −3.95512167224869046953337178271, −3.29806824654676821949101014722, −2.37938682626001845996927010533, −0.57210283453995854727359200788,
0.57210283453995854727359200788, 2.37938682626001845996927010533, 3.29806824654676821949101014722, 3.95512167224869046953337178271, 4.39270631729122541929696421422, 5.11246027075057276079437217649, 6.43905895974355756267404135720, 6.84673778174044601773381981508, 7.74544652508325868715628394179, 8.563544304455625275261764919554