| L(s) = 1 | − 13.4i·3-s + 138. i·7-s + 62.9·9-s + 259.·11-s − 154i·13-s + 178i·17-s − 965.·19-s + 1.86e3·21-s − 2.63e3i·23-s − 4.10e3i·27-s − 4.11e3·29-s + 3.15e3·31-s − 3.48e3i·33-s + 7.44e3i·37-s − 2.06e3·39-s + ⋯ |
| L(s) = 1 | − 0.860i·3-s + 1.06i·7-s + 0.259·9-s + 0.646·11-s − 0.252i·13-s + 0.149i·17-s − 0.613·19-s + 0.920·21-s − 1.03i·23-s − 1.08i·27-s − 0.907·29-s + 0.590·31-s − 0.556i·33-s + 0.893i·37-s − 0.217·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.366805367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366805367\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 13.4iT - 243T^{2} \) |
| 7 | \( 1 - 138. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 259.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 154iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 178iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 965.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.63e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.44e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.27e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.79e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 7.41e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.22e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.98e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.85e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.86e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.08e5iT - 8.58e9T^{2} \) |
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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
−9.353973373998657656542316018854, −8.481012887961840846287171249618, −7.82663758824097923669667712184, −6.65553626169735182784301470583, −6.24403911066181939868423854822, −5.09478022437639887587343039248, −3.99412607673623166964252787070, −2.64575071258979439869286728521, −1.82674862289652140693352811158, −0.71962466511199404234081476054,
0.71153054387486928716542038399, 1.88403841735864449719640416385, 3.52270301494093885297101614858, 4.06574541267715744060213379862, 4.88539348994232004145439305586, 6.06417016047626141266947412115, 7.09856804371064434662156661704, 7.72885481782015995278886808443, 9.097411148008982936456625408406, 9.490538319044207041817344594487
