| L(s) = 1 | − 0.712i·3-s + 7.86·7-s + 8.49·9-s + (2.47 − 10.7i)11-s − 1.76·13-s + 15.1·17-s + 3.61i·19-s − 5.60i·21-s − 10.7i·23-s − 12.4i·27-s + 26.5i·29-s − 21.3·31-s + (−7.64 − 1.76i)33-s + 1.48i·37-s + 1.25i·39-s + ⋯ |
| L(s) = 1 | − 0.237i·3-s + 1.12·7-s + 0.943·9-s + (0.225 − 0.974i)11-s − 0.135·13-s + 0.888·17-s + 0.190i·19-s − 0.267i·21-s − 0.465i·23-s − 0.461i·27-s + 0.916i·29-s − 0.690·31-s + (−0.231 − 0.0535i)33-s + 0.0400i·37-s + 0.0322i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.508556322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508556322\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-2.47 + 10.7i)T \) |
| good | 3 | \( 1 + 0.712iT - 9T^{2} \) |
| 7 | \( 1 - 7.86T + 49T^{2} \) |
| 13 | \( 1 + 1.76T + 169T^{2} \) |
| 17 | \( 1 - 15.1T + 289T^{2} \) |
| 19 | \( 1 - 3.61iT - 361T^{2} \) |
| 23 | \( 1 + 10.7iT - 529T^{2} \) |
| 29 | \( 1 - 26.5iT - 841T^{2} \) |
| 31 | \( 1 + 21.3T + 961T^{2} \) |
| 37 | \( 1 - 1.48iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 23.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 1.97iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 78.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 27.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 56.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 41.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 106.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 89.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 132.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 59.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 46.9iT - 9.40e3T^{2} \) |
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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
−9.514082862669359637909772481259, −8.673327782906429320372201393090, −7.84859377271407765558839672663, −7.25370809159610640124178342639, −6.16415235181517737780022312408, −5.26591688632768735437872773248, −4.35752503890098390545098371579, −3.32579467226131496545648061818, −1.89477873958005865076929166361, −0.927050725156500011089994237956,
1.22100286480387013261055872595, 2.18392715172282912971795909199, 3.72888449567890079487112621898, 4.58049721040316059042233306097, 5.24215443913506699671638950554, 6.43484766532486597535554265995, 7.58408553499127233042598671608, 7.77210480785331916420370782132, 9.119592511031219321946139182145, 9.723379000271423027933650993142
