L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.173 + 0.300i)3-s + (−0.499 − 0.866i)4-s + (−0.173 − 0.300i)6-s + 0.999·8-s + (0.439 + 0.761i)9-s + 1.53·11-s + 0.347·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 0.879·18-s + (−0.766 + 1.32i)22-s + (−0.173 + 0.300i)24-s + (−0.5 − 0.866i)25-s − 0.652·27-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.173 + 0.300i)3-s + (−0.499 − 0.866i)4-s + (−0.173 − 0.300i)6-s + 0.999·8-s + (0.439 + 0.761i)9-s + 1.53·11-s + 0.347·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s − 0.879·18-s + (−0.766 + 1.32i)22-s + (−0.173 + 0.300i)24-s + (−0.5 − 0.866i)25-s − 0.652·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9988656352\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9988656352\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.53T + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 1.87T + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.996450550409292916188266202610, −8.430638847626936748346153350041, −7.27682866423810615815885230601, −7.16769427035594273135207871221, −5.99425158177577190382029920822, −5.47783112961325386666246636624, −4.42397315127002355419516279147, −3.94815570727009404188752212725, −2.28733973991170197725997292149, −1.07526277125942176532524721440,
1.09733123894287451397950762083, 1.78165928957934329060036778495, 3.21606771023013407189258209557, 3.85757521192815391971078139964, 4.58768929848599925020705763823, 5.94603380500026527013267250421, 6.57971796465804316119024589595, 7.46836445295394718609899827262, 8.119581355924151838569181100761, 9.179900369418957763537576264330