| L(s) = 1 | + 1.73·3-s − 6.92i·5-s + 12i·7-s + 2.99·9-s + 6.92·11-s + 13.8i·13-s − 11.9i·15-s + 14·17-s + 34.6·19-s + 20.7i·21-s − 24i·23-s − 22.9·25-s + 5.19·27-s − 34.6i·29-s − 12i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.38i·5-s + 1.71i·7-s + 0.333·9-s + 0.629·11-s + 1.06i·13-s − 0.799i·15-s + 0.823·17-s + 1.82·19-s + 0.989i·21-s − 1.04i·23-s − 0.919·25-s + 0.192·27-s − 1.19i·29-s − 0.387i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.23040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23040\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| good | 5 | \( 1 + 6.92iT - 25T^{2} \) |
| 7 | \( 1 - 12iT - 49T^{2} \) |
| 11 | \( 1 - 6.92T + 121T^{2} \) |
| 13 | \( 1 - 13.8iT - 169T^{2} \) |
| 17 | \( 1 - 14T + 289T^{2} \) |
| 19 | \( 1 - 34.6T + 361T^{2} \) |
| 23 | \( 1 + 24iT - 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 + 12iT - 961T^{2} \) |
| 37 | \( 1 + 27.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 14T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.92T + 1.84e3T^{2} \) |
| 47 | \( 1 - 72iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 62.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 48.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 55.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 90.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 24iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 50T + 5.32e3T^{2} \) |
| 79 | \( 1 + 12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 20.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 62T + 7.92e3T^{2} \) |
| 97 | \( 1 + 146T + 9.40e3T^{2} \) |
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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
−11.51661705814226703957888376779, −9.652334245406129556665629293816, −9.236745256398638902130796866753, −8.596098813989652882076652746726, −7.65877302346149446504204491785, −6.13785766961229702628623374441, −5.25072535788889615179863866723, −4.20552734000691682364851634956, −2.66824947096284882253026439140, −1.34286285132347209182417282391,
1.21046312263339819291986840948, 3.32618197500386366211747823630, 3.48388600920688756217473604560, 5.24723097026686343974978511357, 6.77952368631480579094103210185, 7.30997158540891970705700473266, 8.025490629098296322970742994711, 9.631497547297520069758279184686, 10.19350955661263956310695754507, 10.91247686545957982044319916687
