L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 1.35·7-s + 8-s + 9-s + 10-s − 3.19·11-s − 12-s − 1.35·14-s − 15-s + 16-s − 2.04·17-s + 18-s + 5.34·19-s + 20-s + 1.35·21-s − 3.19·22-s − 8.32·23-s − 24-s + 25-s − 27-s − 1.35·28-s + 8.07·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.512·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.964·11-s − 0.288·12-s − 0.362·14-s − 0.258·15-s + 0.250·16-s − 0.496·17-s + 0.235·18-s + 1.22·19-s + 0.223·20-s + 0.296·21-s − 0.681·22-s − 1.73·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.256·28-s + 1.49·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 3.19T + 11T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 8.32T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 - 0.185T + 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 - 1.62T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 - 2.32T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 + 1.42T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 - 4.74T + 97T^{2} \) |
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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
−7.85339447891466894066862859484, −6.69929234653148672318163086771, −6.53138482965282130209282598614, −5.49309005430688837689754879771, −5.15224090714894771576650984062, −4.25667898056802554360203127778, −3.30062793265430299673585867490, −2.53608696220949623574023388777, −1.49268311438731401837633331249, 0,
1.49268311438731401837633331249, 2.53608696220949623574023388777, 3.30062793265430299673585867490, 4.25667898056802554360203127778, 5.15224090714894771576650984062, 5.49309005430688837689754879771, 6.53138482965282130209282598614, 6.69929234653148672318163086771, 7.85339447891466894066862859484