L(s) = 1 | − 2-s − 3-s + 4-s − 2.45·5-s + 6-s − 4.70·7-s − 8-s + 9-s + 2.45·10-s + 1.91·11-s − 12-s − 5.90·13-s + 4.70·14-s + 2.45·15-s + 16-s − 1.14·17-s − 18-s − 19-s − 2.45·20-s + 4.70·21-s − 1.91·22-s + 3.30·23-s + 24-s + 1.04·25-s + 5.90·26-s − 27-s − 4.70·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.09·5-s + 0.408·6-s − 1.77·7-s − 0.353·8-s + 0.333·9-s + 0.777·10-s + 0.575·11-s − 0.288·12-s − 1.63·13-s + 1.25·14-s + 0.635·15-s + 0.250·16-s − 0.277·17-s − 0.235·18-s − 0.229·19-s − 0.549·20-s + 1.02·21-s − 0.407·22-s + 0.689·23-s + 0.204·24-s + 0.209·25-s + 1.15·26-s − 0.192·27-s − 0.888·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + 5.90T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 23 | \( 1 - 3.30T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 - 7.67T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 - 6.19T + 41T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 - 5.78T + 47T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 3.23T + 67T^{2} \) |
| 71 | \( 1 - 1.22T + 71T^{2} \) |
| 73 | \( 1 - 7.85T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 5.90T + 89T^{2} \) |
| 97 | \( 1 + 0.0999T + 97T^{2} \) |
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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
−7.61262924924444485304233765218, −6.85598902589973812856495395179, −6.72232141254836034303145325060, −5.77588449713177141354507013608, −4.76709574892509540961070757822, −3.99009705631227732170649182659, −3.15744680209677214399698897151, −2.39801207024205425817159112332, −0.77803566384066747937630543196, 0,
0.77803566384066747937630543196, 2.39801207024205425817159112332, 3.15744680209677214399698897151, 3.99009705631227732170649182659, 4.76709574892509540961070757822, 5.77588449713177141354507013608, 6.72232141254836034303145325060, 6.85598902589973812856495395179, 7.61262924924444485304233765218