L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 4·7-s − 8-s + 9-s − 10-s − 12-s − 13-s − 4·14-s − 15-s + 16-s − 6·17-s − 18-s + 8·19-s + 20-s − 4·21-s + 4·23-s + 24-s + 25-s + 26-s − 27-s + 4·28-s − 2·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.872·21-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.12511533699244, −14.26635058298004, −13.84461609585650, −13.41228189980767, −12.69639235262941, −12.00128537438596, −11.57418237322208, −11.30392209441165, −10.67322909632018, −10.27854259863895, −9.628088865510703, −9.048286558761366, −8.596912068762103, −8.048122017415341, −7.316236252620702, −7.006145479420949, −6.436861608145082, −5.484282468464505, −5.170386163825680, −4.779120865889971, −3.861580489045557, −3.047346988320038, −2.178831242089690, −1.628859100067623, −1.057152906345285, 0,
1.057152906345285, 1.628859100067623, 2.178831242089690, 3.047346988320038, 3.861580489045557, 4.779120865889971, 5.170386163825680, 5.484282468464505, 6.436861608145082, 7.006145479420949, 7.316236252620702, 8.048122017415341, 8.596912068762103, 9.048286558761366, 9.628088865510703, 10.27854259863895, 10.67322909632018, 11.30392209441165, 11.57418237322208, 12.00128537438596, 12.69639235262941, 13.41228189980767, 13.84461609585650, 14.26635058298004, 15.12511533699244