L(s) = 1 | + (−1 + 1.73i)5-s + (2 + 1.73i)7-s + (1 + 1.73i)11-s + (1.5 + 2.59i)13-s + (4 − 6.92i)17-s + (0.5 + 0.866i)19-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s + (2 − 3.46i)29-s + 3·31-s + (−5 + 1.73i)35-s + (0.5 + 0.866i)37-s + (3 + 5.19i)41-s + (−5.5 + 9.52i)43-s − 6·47-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s + (0.755 + 0.654i)7-s + (0.301 + 0.522i)11-s + (0.416 + 0.720i)13-s + (0.970 − 1.68i)17-s + (0.114 + 0.198i)19-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s + (0.371 − 0.643i)29-s + 0.538·31-s + (−0.845 + 0.292i)35-s + (0.0821 + 0.142i)37-s + (0.468 + 0.811i)41-s + (−0.838 + 1.45i)43-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.946653573\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.946653573\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-4 + 6.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + (-1 + 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.285702117274318826432183164128, −8.242043452222187806673227761602, −7.69023672038116709112585594124, −6.80859673526925755125187882360, −6.22948151303062848260184101696, −4.96912370270836326265886419669, −4.53217634924388527413796181093, −3.21140986362217815392390639253, −2.53850821728945269921201718714, −1.20498453161441346019652576867,
0.826001396944456149174647900534, 1.61824649618290751723304394337, 3.38796596737608600423443744943, 3.86709123919351988825222384290, 4.98893842650271006639456741843, 5.52621592511868185873439538250, 6.58006057598810795509374823461, 7.53246245241911807888563446648, 8.331914229154917331808340038655, 8.484636550045670076941386799302