| L(s) = 1 | + (−0.409 − 0.236i)2-s + (−0.888 − 1.53i)4-s + (2.21 + 1.28i)7-s + 1.78i·8-s + (−3.08 + 5.34i)11-s + (1.84 − 1.06i)13-s + (−0.606 − 1.05i)14-s + (−1.35 + 2.34i)16-s + 3.16i·17-s − 0.356·19-s + (2.52 − 1.45i)22-s + (3.64 − 2.10i)23-s − 1.00·26-s − 4.55i·28-s + (−0.843 + 1.46i)29-s + ⋯ |
| L(s) = 1 | + (−0.289 − 0.167i)2-s + (−0.444 − 0.769i)4-s + (0.838 + 0.484i)7-s + 0.631i·8-s + (−0.929 + 1.61i)11-s + (0.512 − 0.295i)13-s + (−0.162 − 0.280i)14-s + (−0.338 + 0.585i)16-s + 0.768i·17-s − 0.0817·19-s + (0.539 − 0.311i)22-s + (0.760 − 0.439i)23-s − 0.197·26-s − 0.860i·28-s + (−0.156 + 0.271i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05464 + 0.331534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05464 + 0.331534i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.409 + 0.236i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-2.21 - 1.28i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.08 - 5.34i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.84 + 1.06i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 0.356T + 19T^{2} \) |
| 23 | \( 1 + (-3.64 + 2.10i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.843 - 1.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.12 - 7.15i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.63iT - 37T^{2} \) |
| 41 | \( 1 + (1.36 + 2.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.64 - 3.83i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-9.89 - 5.71i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 9.43iT - 53T^{2} \) |
| 59 | \( 1 + (5.10 + 8.84i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.00549 + 0.00952i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.851 - 0.491i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 - 6.61iT - 73T^{2} \) |
| 79 | \( 1 + (4.73 - 8.20i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.02 + 5.20i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + (-6.24 - 3.60i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.62341590095493629309378364697, −9.792422986873194895512048526262, −8.865668983837268262470523792343, −8.181540113179742016670757744610, −7.18121030666380912032869026143, −5.91489559416634201155482772799, −5.06600040855855107544386239094, −4.37582795804762879343174877040, −2.51012044958995891946227828752, −1.42786104377166248859905433447,
0.72622528814150035199321171453, 2.77268394602588955640819559035, 3.82994904404128646452368406275, 4.85572445257730771382155631660, 5.90788080841766085756222547738, 7.18785871199132767185116302337, 7.933939849155877640268532348427, 8.535502936785727408784499031005, 9.336336039060065700314875042599, 10.48673438774778454798245216546
