L(s) = 1 | + (0.274 − 0.158i)3-s + (−4.91 − 2.83i)5-s + (−4.44 + 7.70i)9-s + (−7.94 − 13.7i)11-s + 20.1i·13-s − 1.79·15-s + (−0.823 + 0.475i)17-s + (27.5 + 15.9i)19-s + (13 − 22.5i)23-s + (3.60 + 6.23i)25-s + 5.67i·27-s + 27.7·29-s + (13.1 − 7.57i)31-s + (−4.36 − 2.52i)33-s + (16 − 27.7i)37-s + ⋯ |
L(s) = 1 | + (0.0915 − 0.0528i)3-s + (−0.982 − 0.567i)5-s + (−0.494 + 0.856i)9-s + (−0.722 − 1.25i)11-s + 1.55i·13-s − 0.119·15-s + (−0.0484 + 0.0279i)17-s + (1.45 + 0.838i)19-s + (0.565 − 0.978i)23-s + (0.144 + 0.249i)25-s + 0.210i·27-s + 0.958·29-s + (0.423 − 0.244i)31-s + (−0.132 − 0.0763i)33-s + (0.432 − 0.748i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.345845084\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.345845084\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.274 + 0.158i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (4.91 + 2.83i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (7.94 + 13.7i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 20.1iT - 169T^{2} \) |
| 17 | \( 1 + (0.823 - 0.475i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-27.5 - 15.9i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-13 + 22.5i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 27.7T + 841T^{2} \) |
| 31 | \( 1 + (-13.1 + 7.57i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-16 + 27.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 17.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 59.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (66.1 + 38.1i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (12.8 + 22.3i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-59.2 + 34.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-46.9 - 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-43.6 - 75.6i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-60.9 + 35.1i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-20.1 + 34.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 71.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-66.9 - 38.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 128. iT - 9.40e3T^{2} \) |
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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
−10.11808836507667222033293195789, −9.002250170720048906528280267070, −8.267583636225737231803665702191, −7.80547533782298240909852469597, −6.64640337191632170097296426277, −5.50061422890738784000099531243, −4.66351679842971467933754230282, −3.63192900556014054800762951165, −2.45696730720666251029323661770, −0.76719653219111930018285321991,
0.73411706587366073794026647819, 2.83783135644667312846498871826, 3.34590158841477662585361260819, 4.69617304571454689295175524458, 5.58389741997667747968888949949, 6.86435150072554999544148506009, 7.55666931668939901477635275889, 8.178068966738756215175565104465, 9.399972270587160605584759740228, 10.05258612550999600483838359574