| L(s) = 1 | + (2.5 − 10.8i)5-s − 8.71i·7-s + 11·11-s − 69.7i·13-s + 26.1i·17-s + 68·19-s + 117. i·23-s + (−112. − 54.4i)25-s + 260·29-s + 175·31-s + (−95.0 − 21.7i)35-s − 169. i·37-s + 380·41-s − 305. i·43-s − 305. i·47-s + ⋯ |
| L(s) = 1 | + (0.223 − 0.974i)5-s − 0.470i·7-s + 0.301·11-s − 1.48i·13-s + 0.373i·17-s + 0.821·19-s + 1.06i·23-s + (−0.900 − 0.435i)25-s + 1.66·29-s + 1.01·31-s + (−0.458 − 0.105i)35-s − 0.755i·37-s + 1.44·41-s − 1.08i·43-s − 0.946i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.483316685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483316685\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 + 10.8i)T \) |
| 11 | \( 1 - 11T \) |
| good | 7 | \( 1 + 8.71iT - 343T^{2} \) |
| 13 | \( 1 + 69.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 26.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 68T + 6.85e3T^{2} \) |
| 23 | \( 1 - 117. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 260T + 2.43e4T^{2} \) |
| 31 | \( 1 - 175T + 2.97e4T^{2} \) |
| 37 | \( 1 + 169. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 380T + 6.89e4T^{2} \) |
| 43 | \( 1 + 305. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 305. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 453. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 143T + 2.05e5T^{2} \) |
| 61 | \( 1 - 676T + 2.26e5T^{2} \) |
| 67 | \( 1 - 527. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 331. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 218T + 4.93e5T^{2} \) |
| 83 | \( 1 + 758. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 771. iT - 9.12e5T^{2} \) |
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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
−8.574854886794427360538462551706, −7.86527760275028967830721689350, −7.18254850951836399782009354714, −5.96897751045701339981543214374, −5.45515807293709954606030001929, −4.53714247563273278842135516383, −3.67209549887555450840721915902, −2.61644326806454134236334441125, −1.22577821697174841978585500194, −0.61601196103267259630722952755,
1.09166892774303373528788370038, 2.35472722028641628037614299963, 2.95712583691776375464175823487, 4.16412642221669778699775583983, 4.92412690831837567722923623385, 6.20195781422043926063889723444, 6.51870855821731923322130719842, 7.34836172403514458929868958015, 8.288955118672859731906101475733, 9.122981395325553270234618982409
