L(s) = 1 | + 1.41·2-s + 1.41·3-s + 1.41·5-s + 2.00·6-s − 7-s − 2.82·8-s − 0.999·9-s + 2.00·10-s − 4.82·11-s + 6.82·13-s − 1.41·14-s + 2.00·15-s − 4.00·16-s − 6.65·17-s − 1.41·18-s + 6.65·19-s − 1.41·21-s − 6.82·22-s − 3.17·23-s − 4·24-s − 2.99·25-s + 9.65·26-s − 5.65·27-s − 6.82·29-s + 2.82·30-s + ⋯ |
L(s) = 1 | + 1.00·2-s + 0.816·3-s + 0.632·5-s + 0.816·6-s − 0.377·7-s − 0.999·8-s − 0.333·9-s + 0.632·10-s − 1.45·11-s + 1.89·13-s − 0.377·14-s + 0.516·15-s − 1.00·16-s − 1.61·17-s − 0.333·18-s + 1.52·19-s − 0.308·21-s − 1.45·22-s − 0.661·23-s − 0.816·24-s − 0.599·25-s + 1.89·26-s − 1.08·27-s − 1.26·29-s + 0.516·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4003 | \( 1+O(T) \) |
good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 6.82T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 + 9.82T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 6.17T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.151364239265583733926626653248, −7.38015678836174200313665146198, −6.26011756104634778170814493927, −5.70019151937449400557254896563, −5.26053723403507263980916319769, −4.01290259266816417020876944909, −3.50585943082748260518053743449, −2.71798638211430985859676953885, −1.90092769902051903625201955119, 0,
1.90092769902051903625201955119, 2.71798638211430985859676953885, 3.50585943082748260518053743449, 4.01290259266816417020876944909, 5.26053723403507263980916319769, 5.70019151937449400557254896563, 6.26011756104634778170814493927, 7.38015678836174200313665146198, 8.151364239265583733926626653248