L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 4·7-s + 8-s + 9-s + 2·10-s − 4·11-s − 12-s − 4·14-s − 2·15-s + 16-s − 4·17-s + 18-s + 2·20-s + 4·21-s − 4·22-s − 4·23-s − 24-s − 25-s − 27-s − 4·28-s + 2·29-s − 2·30-s − 4·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s − 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.447·20-s + 0.872·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.755·28-s + 0.371·29-s − 0.365·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.85183322394734, −12.45469966106728, −11.96164310530783, −11.38421100694646, −10.97608506709356, −10.45635863500379, −10.04514219419014, −9.734062549614414, −9.362473496338470, −8.687552149172947, −8.026245872932860, −7.641980180196087, −6.941077346754210, −6.512579123134512, −6.189082450562824, −5.898712074317652, −5.262902703548281, −4.858864187925205, −4.372943118711261, −3.604295990615784, −3.230841415242928, −2.645224551182918, −2.081506149968751, −1.681941640916114, −0.5723533801974151, 0,
0.5723533801974151, 1.681941640916114, 2.081506149968751, 2.645224551182918, 3.230841415242928, 3.604295990615784, 4.372943118711261, 4.858864187925205, 5.262902703548281, 5.898712074317652, 6.189082450562824, 6.512579123134512, 6.941077346754210, 7.641980180196087, 8.026245872932860, 8.687552149172947, 9.362473496338470, 9.734062549614414, 10.04514219419014, 10.45635863500379, 10.97608506709356, 11.38421100694646, 11.96164310530783, 12.45469966106728, 12.85183322394734