L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s − 5·11-s − 12-s + 2·14-s − 15-s + 16-s + 5·17-s − 18-s + 5·19-s + 20-s + 2·21-s + 5·22-s − 4·23-s + 24-s + 25-s − 27-s − 2·28-s − 4·29-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 − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.436·21-s + 1.06·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 0.742·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.66295604682882, −14.17029582877676, −13.45590313742270, −13.14168186414777, −12.56070422293977, −12.03890282725504, −11.63416102062028, −10.87869008531434, −10.47090937539543, −9.956115373225726, −9.678360990635782, −9.195042805949031, −8.240556449018275, −7.917487866353255, −7.349240285705038, −6.852551846835075, −6.069871958560200, −5.677068018685665, −5.287449199892859, −4.542711225745672, −3.522898691208819, −3.082340498484024, −2.387693446925775, −1.602686878566146, −0.7744770195358431, 0,
0.7744770195358431, 1.602686878566146, 2.387693446925775, 3.082340498484024, 3.522898691208819, 4.542711225745672, 5.287449199892859, 5.677068018685665, 6.069871958560200, 6.852551846835075, 7.349240285705038, 7.917487866353255, 8.240556449018275, 9.195042805949031, 9.678360990635782, 9.956115373225726, 10.47090937539543, 10.87869008531434, 11.63416102062028, 12.03890282725504, 12.56070422293977, 13.14168186414777, 13.45590313742270, 14.17029582877676, 14.66295604682882