L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 12-s − 2·14-s − 15-s + 16-s + 18-s − 2·19-s − 20-s − 2·21-s − 6·23-s + 24-s + 25-s + 27-s − 2·28-s − 30-s − 8·31-s + 32-s + 2·35-s + 36-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 + 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.182·30-s − 1.43·31-s + 0.176·32-s + 0.338·35-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.88593326090765133388802736853, −7.01063795361233220115271892259, −6.52612704147769182478515055757, −5.65684246699572527440407796386, −4.84328518077824642071614416841, −3.85688187562639441403286843942, −3.54548364587982975263633598948, −2.56884112905537523152532473910, −1.68073902851975012054345242127, 0,
1.68073902851975012054345242127, 2.56884112905537523152532473910, 3.54548364587982975263633598948, 3.85688187562639441403286843942, 4.84328518077824642071614416841, 5.65684246699572527440407796386, 6.52612704147769182478515055757, 7.01063795361233220115271892259, 7.88593326090765133388802736853