L(s) = 1 | − 2.58·2-s + 0.285·3-s + 4.70·4-s − 3.09·5-s − 0.738·6-s + 5.20·7-s − 6.99·8-s − 2.91·9-s + 8.00·10-s + 2.28·11-s + 1.34·12-s + 1.50·13-s − 13.4·14-s − 0.881·15-s + 8.70·16-s − 1.74·17-s + 7.55·18-s − 19-s − 14.5·20-s + 1.48·21-s − 5.90·22-s − 1.32·23-s − 1.99·24-s + 4.56·25-s − 3.90·26-s − 1.68·27-s + 24.4·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 0.164·3-s + 2.35·4-s − 1.38·5-s − 0.301·6-s + 1.96·7-s − 2.47·8-s − 0.972·9-s + 2.53·10-s + 0.688·11-s + 0.386·12-s + 0.418·13-s − 3.59·14-s − 0.227·15-s + 2.17·16-s − 0.423·17-s + 1.78·18-s − 0.229·19-s − 3.25·20-s + 0.323·21-s − 1.25·22-s − 0.277·23-s − 0.407·24-s + 0.912·25-s − 0.765·26-s − 0.324·27-s + 4.62·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 - 0.285T + 3T^{2} \) |
| 5 | \( 1 + 3.09T + 5T^{2} \) |
| 7 | \( 1 - 5.20T + 7T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 - 4.11T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + 7.87T + 41T^{2} \) |
| 43 | \( 1 + 6.35T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 1.36T + 53T^{2} \) |
| 59 | \( 1 - 2.27T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 1.39T + 67T^{2} \) |
| 71 | \( 1 - 3.32T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 9.38T + 79T^{2} \) |
| 83 | \( 1 + 6.42T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.197551129566521205220836704614, −7.85889758870656734165189748706, −7.01620040454137213879143829178, −6.25204008607328122431802839959, −5.06672016786868591048623363652, −4.18664797167709801082134759235, −3.14693061735681974816943192872, −2.01835462742641236870487251934, −1.21394206617955529530228342928, 0,
1.21394206617955529530228342928, 2.01835462742641236870487251934, 3.14693061735681974816943192872, 4.18664797167709801082134759235, 5.06672016786868591048623363652, 6.25204008607328122431802839959, 7.01620040454137213879143829178, 7.85889758870656734165189748706, 8.197551129566521205220836704614