| L(s) = 1 | − 2i·7-s − 4i·11-s + 6i·17-s − 4i·19-s + 4i·23-s + 6i·29-s − 10·31-s + 4·37-s − 10·41-s − 4·43-s − 4i·47-s + 3·49-s − 10·53-s + 8i·59-s + 8i·61-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s − 1.20i·11-s + 1.45i·17-s − 0.917i·19-s + 0.834i·23-s + 1.11i·29-s − 1.79·31-s + 0.657·37-s − 1.56·41-s − 0.609·43-s − 0.583i·47-s + 0.428·49-s − 1.37·53-s + 1.04i·59-s + 1.02i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6679904337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6679904337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.174724527848889956806089709828, −7.37275693547771654853789522853, −6.82572682119283573777787240605, −5.99489350071448969992445769950, −5.41136457983907222899504045179, −4.54387220811817044759358299572, −3.57631236054534981209029571192, −3.30975093088840062450932444575, −1.94932438853790802435305219186, −1.07037968262748505843695831373,
0.16654237439607242280517030264, 1.70213899085251737986528195714, 2.35321444672233488719118895608, 3.24108789999648633850882728161, 4.19761845725705193880022237990, 4.94693218148454056287421844696, 5.50049250889242634962875459496, 6.38453034787917308936478963289, 7.01524480659532236746056979592, 7.72245141839560348990795671873
