L(s) = 1 | − 1.61·2-s − 3-s + 0.621·4-s + 1.62·5-s + 1.61·6-s + 3.15·7-s + 2.23·8-s + 9-s − 2.62·10-s − 2.23·11-s − 0.621·12-s − 1.49·13-s − 5.11·14-s − 1.62·15-s − 4.85·16-s − 17-s − 1.61·18-s − 3.38·19-s + 1.00·20-s − 3.15·21-s + 3.61·22-s + 7.59·23-s − 2.23·24-s − 2.37·25-s + 2.41·26-s − 27-s + 1.96·28-s + ⋯ |
L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.310·4-s + 0.725·5-s + 0.661·6-s + 1.19·7-s + 0.789·8-s + 0.333·9-s − 0.830·10-s − 0.673·11-s − 0.179·12-s − 0.413·13-s − 1.36·14-s − 0.418·15-s − 1.21·16-s − 0.242·17-s − 0.381·18-s − 0.776·19-s + 0.225·20-s − 0.689·21-s + 0.770·22-s + 1.58·23-s − 0.455·24-s − 0.474·25-s + 0.473·26-s − 0.192·27-s + 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8959914985\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8959914985\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 - 7.59T + 23T^{2} \) |
| 29 | \( 1 + 6.67T + 29T^{2} \) |
| 31 | \( 1 + 2.84T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 8.52T + 47T^{2} \) |
| 53 | \( 1 - 7.27T + 53T^{2} \) |
| 59 | \( 1 - 1.82T + 59T^{2} \) |
| 61 | \( 1 - 1.27T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 3.88T + 97T^{2} \) |
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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
−8.485266678919941794948036413293, −7.74577020595522599039624050124, −7.28879016628730793239122958574, −6.32873933239224054726158375174, −5.35574957794584454943266645113, −4.92063100976056147467369579492, −4.05444731715266286464066647692, −2.40135580234800064849155376442, −1.75381687133843549934283851821, −0.68134204419067878525433273602,
0.68134204419067878525433273602, 1.75381687133843549934283851821, 2.40135580234800064849155376442, 4.05444731715266286464066647692, 4.92063100976056147467369579492, 5.35574957794584454943266645113, 6.32873933239224054726158375174, 7.28879016628730793239122958574, 7.74577020595522599039624050124, 8.485266678919941794948036413293