L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 2·13-s − 15-s − 6·17-s + 4·19-s − 21-s + 23-s + 25-s − 27-s − 2·29-s + 4·31-s + 35-s − 6·37-s − 2·39-s + 2·41-s − 4·43-s + 45-s + 49-s + 6·51-s + 2·53-s − 4·57-s + 8·59-s − 2·61-s + 63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.169·35-s − 0.986·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.840·51-s + 0.274·53-s − 0.529·57-s + 1.04·59-s − 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.20305045675713, −14.58390689843002, −13.84557331753141, −13.57445901162406, −13.08881450864427, −12.46265789795032, −11.86496848724599, −11.32937670849682, −11.04395133200080, −10.32067422005249, −9.957049653431983, −9.177008146824321, −8.770091677133429, −8.204981264898562, −7.411090628210387, −6.910482519531531, −6.390033337717169, −5.750011286282166, −5.245331252888869, −4.629529194233612, −4.060149377631481, −3.253783471800847, −2.468771814557569, −1.711379580956826, −1.051224724891664, 0,
1.051224724891664, 1.711379580956826, 2.468771814557569, 3.253783471800847, 4.060149377631481, 4.629529194233612, 5.245331252888869, 5.750011286282166, 6.390033337717169, 6.910482519531531, 7.411090628210387, 8.204981264898562, 8.770091677133429, 9.177008146824321, 9.957049653431983, 10.32067422005249, 11.04395133200080, 11.32937670849682, 11.86496848724599, 12.46265789795032, 13.08881450864427, 13.57445901162406, 13.84557331753141, 14.58390689843002, 15.20305045675713