L(s) = 1 | + (−0.618 + 1.61i)3-s + (2.61 + 0.381i)7-s + (−2.23 − 2.00i)9-s + 0.763i·11-s − 1.23·13-s − 4.47i·17-s − 7.23i·19-s + (−2.23 + 4i)21-s + 7.70·23-s + (4.61 − 2.38i)27-s + 4i·29-s − 3.23i·31-s + (−1.23 − 0.472i)33-s − 4.47i·37-s + (0.763 − 2.00i)39-s + ⋯ |
L(s) = 1 | + (−0.356 + 0.934i)3-s + (0.989 + 0.144i)7-s + (−0.745 − 0.666i)9-s + 0.230i·11-s − 0.342·13-s − 1.08i·17-s − 1.66i·19-s + (−0.487 + 0.872i)21-s + 1.60·23-s + (0.888 − 0.458i)27-s + 0.742i·29-s − 0.581i·31-s + (−0.215 − 0.0821i)33-s − 0.735i·37-s + (0.122 − 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.591721316\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.591721316\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 - 1.61i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.61 - 0.381i)T \) |
good | 11 | \( 1 - 0.763iT - 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 7.23iT - 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 3.23iT - 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 7.23iT - 43T^{2} \) |
| 47 | \( 1 - 3.23iT - 47T^{2} \) |
| 53 | \( 1 - 9.23T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 4.94iT - 61T^{2} \) |
| 67 | \( 1 + 9.70iT - 67T^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 6.76T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.23iT - 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−9.067469549302979535867504025351, −8.659514095389717245688216628788, −7.39660204046509146873574651152, −6.89165844418245535756177615017, −5.61811748082111025146344498183, −4.92048213135829559113682701064, −4.56587738643557640763158859376, −3.27875113904493878980602975206, −2.36118100196458779100390349741, −0.68942749501557795705789754907,
1.13859493635226837043265970847, 1.91293169771414354389330850880, 3.14290411082387275665072343726, 4.34892359100562789941374675700, 5.27064777221929686863335525495, 5.95608630033933131489738902268, 6.80577973365853109344000948248, 7.62898531030116609705594530414, 8.214227446497854548101518457587, 8.781917117972580118004983619562