| L(s) = 1 | + 19.5i·3-s + 35.0i·7-s − 138.·9-s + 426.·11-s + 1.10e3i·13-s − 109. i·17-s + 495.·19-s − 684.·21-s − 2.49e3i·23-s + 2.04e3i·27-s + 42.4·29-s + 7.99e3·31-s + 8.31e3i·33-s + 1.37e4i·37-s − 2.15e4·39-s + ⋯ |
| L(s) = 1 | + 1.25i·3-s + 0.270i·7-s − 0.568·9-s + 1.06·11-s + 1.81i·13-s − 0.0917i·17-s + 0.315·19-s − 0.338·21-s − 0.984i·23-s + 0.540i·27-s + 0.00936·29-s + 1.49·31-s + 1.32i·33-s + 1.65i·37-s − 2.26·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.109650263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.109650263\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 19.5iT - 243T^{2} \) |
| 7 | \( 1 - 35.0iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 426.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.10e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 109. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 495.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.49e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 42.4T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.37e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.18e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.68e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.30e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.78e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.94e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.61e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.32e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.51e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.60e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.13e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.07e5iT - 8.58e9T^{2} \) |
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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
−10.80305212980045218807809189284, −9.725109820380424213106765716269, −9.282738490715755721777270444833, −8.420850028835741052653917071925, −6.92056930323724152326141421754, −6.11243150780367246229444345206, −4.57507322350901849406156385268, −4.28639793002614092324151207971, −2.92925878728820146068957485401, −1.39210243294453318986926915077,
0.57078583167653498487124373135, 1.36512564553566985843762672332, 2.72809883092505504913273257748, 3.96797720615073122359533515122, 5.51126758930304394086993144781, 6.35190325457508689774859263014, 7.39771546973636272469161199618, 7.898112085574360027679138584563, 9.040441327098644985681936263304, 10.11760989646511271037322229875
