L(s) = 1 | + (−0.275 + 0.158i)5-s + (−1.25 − 0.724i)7-s + (0.548 − 0.949i)11-s + (−1.44 − 2.51i)13-s + 3.46i·17-s + 4.89i·19-s + (−1.41 − 2.44i)23-s + (−2.44 + 4.24i)25-s + (7.89 + 4.56i)29-s + (6.45 − 3.72i)31-s + 0.460·35-s + 4.89·37-s + (8.44 − 4.87i)41-s + (−5.97 − 3.44i)43-s + (4.56 − 7.89i)47-s + ⋯ |
L(s) = 1 | + (−0.123 + 0.0710i)5-s + (−0.474 − 0.273i)7-s + (0.165 − 0.286i)11-s + (−0.402 − 0.696i)13-s + 0.840i·17-s + 1.12i·19-s + (−0.294 − 0.510i)23-s + (−0.489 + 0.848i)25-s + (1.46 + 0.846i)29-s + (1.15 − 0.668i)31-s + 0.0778·35-s + 0.805·37-s + (1.31 − 0.761i)41-s + (−0.911 − 0.526i)43-s + (0.665 − 1.15i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.529784752\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.529784752\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.275 - 0.158i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.25 + 0.724i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.548 + 0.949i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.44 + 2.51i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 + (1.41 + 2.44i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.89 - 4.56i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.45 + 3.72i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 + (-8.44 + 4.87i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.97 + 3.44i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.56 + 7.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4.41iT - 53T^{2} \) |
| 59 | \( 1 + (4.56 + 7.89i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.41 - 2.55i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 + (-1.73 - i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.93 + 13.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 11.9iT - 89T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.680269534675530086323345611609, −8.105664585936655620359457389426, −7.36950403654178511163137566860, −6.42342375845813342972799215303, −5.87809884393728255578093347597, −4.88892642031765497724820768202, −3.88221989776505768417597053072, −3.23063766259757920979569249657, −2.06986520845809008423431705928, −0.68814176970196373639511438961,
0.875844794395444511418811140188, 2.40546758839768964431078061685, 3.01454450547233076225326507478, 4.49645412205389667216528942585, 4.65764351354253291370317415875, 6.08359801725226902573516699985, 6.51334766358992893108072091544, 7.44553658117640530348464207736, 8.116676304984280948544434595804, 9.128208044912606386709983451509