L(s) = 1 | − 2-s + 3-s + 4-s − 3.54·5-s − 6-s + 1.31·7-s − 8-s + 9-s + 3.54·10-s + 3.27·11-s + 12-s + 13-s − 1.31·14-s − 3.54·15-s + 16-s + 2.46·17-s − 18-s + 1.21·19-s − 3.54·20-s + 1.31·21-s − 3.27·22-s − 4.22·23-s − 24-s + 7.59·25-s − 26-s + 27-s + 1.31·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.58·5-s − 0.408·6-s + 0.496·7-s − 0.353·8-s + 0.333·9-s + 1.12·10-s + 0.987·11-s + 0.288·12-s + 0.277·13-s − 0.350·14-s − 0.916·15-s + 0.250·16-s + 0.597·17-s − 0.235·18-s + 0.279·19-s − 0.793·20-s + 0.286·21-s − 0.698·22-s − 0.879·23-s − 0.204·24-s + 1.51·25-s − 0.196·26-s + 0.192·27-s + 0.248·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 3.54T + 5T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 1.21T + 19T^{2} \) |
| 23 | \( 1 + 4.22T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 - 9.66T + 41T^{2} \) |
| 43 | \( 1 + 6.54T + 43T^{2} \) |
| 47 | \( 1 + 8.05T + 47T^{2} \) |
| 53 | \( 1 - 0.0283T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 2.25T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 0.0351T + 71T^{2} \) |
| 73 | \( 1 - 7.18T + 73T^{2} \) |
| 79 | \( 1 - 4.16T + 79T^{2} \) |
| 83 | \( 1 + 8.86T + 83T^{2} \) |
| 89 | \( 1 + 7.73T + 89T^{2} \) |
| 97 | \( 1 - 9.04T + 97T^{2} \) |
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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
−7.76116222987683275457750569748, −7.10866974295657809886244469758, −6.38534766998278485162597184919, −5.37534870821522134478876844506, −4.42375132163757755849075764814, −3.65246169715446293534193073571, −3.37020701756405924477592348853, −2.02802309764199751788894174472, −1.21670443082506356069760358187, 0,
1.21670443082506356069760358187, 2.02802309764199751788894174472, 3.37020701756405924477592348853, 3.65246169715446293534193073571, 4.42375132163757755849075764814, 5.37534870821522134478876844506, 6.38534766998278485162597184919, 7.10866974295657809886244469758, 7.76116222987683275457750569748