L(s) = 1 | − 2-s + 3-s + 4-s + 2.82·5-s − 6-s − 4.82·7-s − 8-s + 9-s − 2.82·10-s − 11-s + 12-s − 6.82·13-s + 4.82·14-s + 2.82·15-s + 16-s + 17-s − 18-s + 2.82·20-s − 4.82·21-s + 22-s − 8.82·23-s − 24-s + 3.00·25-s + 6.82·26-s + 27-s − 4.82·28-s − 6·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.26·5-s − 0.408·6-s − 1.82·7-s − 0.353·8-s + 0.333·9-s − 0.894·10-s − 0.301·11-s + 0.288·12-s − 1.89·13-s + 1.29·14-s + 0.730·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.632·20-s − 1.05·21-s + 0.213·22-s − 1.84·23-s − 0.204·24-s + 0.600·25-s + 1.33·26-s + 0.192·27-s − 0.912·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 5.17T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 9.31T + 97T^{2} \) |
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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
−9.502064278063870945508188006554, −8.997664916411671991408803372545, −7.67173561896153376937857402819, −7.07644174441571102455855433117, −6.12172742890835293299970502097, −5.45786915710115672504966668292, −3.83056422080885833549103623886, −2.66742299311593019383381921117, −2.08233188793547099781069858823, 0,
2.08233188793547099781069858823, 2.66742299311593019383381921117, 3.83056422080885833549103623886, 5.45786915710115672504966668292, 6.12172742890835293299970502097, 7.07644174441571102455855433117, 7.67173561896153376937857402819, 8.997664916411671991408803372545, 9.502064278063870945508188006554