L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 1.38·7-s + 8-s + 9-s + 10-s − 12-s − 4.85·13-s − 1.38·14-s − 15-s + 16-s + 6.47·17-s + 18-s − 4.85·19-s + 20-s + 1.38·21-s − 8.61·23-s − 24-s + 25-s − 4.85·26-s − 27-s − 1.38·28-s + 7.23·29-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 − 0.522·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s − 1.34·13-s − 0.369·14-s − 0.258·15-s + 0.250·16-s + 1.56·17-s + 0.235·18-s − 1.11·19-s + 0.223·20-s + 0.301·21-s − 1.79·23-s − 0.204·24-s + 0.200·25-s − 0.951·26-s − 0.192·27-s − 0.261·28-s + 1.34·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 1.38T + 7T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 + 8.61T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 3.14T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 + 9.32T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 3.52T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.87065914553810917024590194379, −7.36408771339446988748830466419, −6.35587051191708998860383390160, −5.99268462097586925552462769469, −5.16274164365577208151797873229, −4.46985032305551930992093064408, −3.53365866370484679943565569113, −2.58478702253765011863624440915, −1.62623074245142524409813029989, 0,
1.62623074245142524409813029989, 2.58478702253765011863624440915, 3.53365866370484679943565569113, 4.46985032305551930992093064408, 5.16274164365577208151797873229, 5.99268462097586925552462769469, 6.35587051191708998860383390160, 7.36408771339446988748830466419, 7.87065914553810917024590194379