L(s) = 1 | + (0.981 − 0.189i)3-s + (0.959 + 0.281i)4-s + (−0.643 + 1.60i)7-s + (0.928 − 0.371i)9-s + (0.995 + 0.0950i)12-s + (−1.12 − 1.17i)13-s + (0.841 + 0.540i)16-s + (−0.0748 − 1.57i)19-s + (−0.327 + 1.70i)21-s + (−0.723 + 0.690i)25-s + (0.841 − 0.540i)27-s + (−1.07 + 1.36i)28-s + (−0.759 + 0.876i)31-s + (0.995 − 0.0950i)36-s + (0.550 + 1.58i)37-s + ⋯ |
L(s) = 1 | + (0.981 − 0.189i)3-s + (0.959 + 0.281i)4-s + (−0.643 + 1.60i)7-s + (0.928 − 0.371i)9-s + (0.995 + 0.0950i)12-s + (−1.12 − 1.17i)13-s + (0.841 + 0.540i)16-s + (−0.0748 − 1.57i)19-s + (−0.327 + 1.70i)21-s + (−0.723 + 0.690i)25-s + (0.841 − 0.540i)27-s + (−1.07 + 1.36i)28-s + (−0.759 + 0.876i)31-s + (0.995 − 0.0950i)36-s + (0.550 + 1.58i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.636008028\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636008028\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.981 + 0.189i)T \) |
| 397 | \( 1 - T \) |
good | 2 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 5 | \( 1 + (0.723 - 0.690i)T^{2} \) |
| 7 | \( 1 + (0.643 - 1.60i)T + (-0.723 - 0.690i)T^{2} \) |
| 11 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 13 | \( 1 + (1.12 + 1.17i)T + (-0.0475 + 0.998i)T^{2} \) |
| 17 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (0.0748 + 1.57i)T + (-0.995 + 0.0950i)T^{2} \) |
| 23 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 29 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 31 | \( 1 + (0.759 - 0.876i)T + (-0.142 - 0.989i)T^{2} \) |
| 37 | \( 1 + (-0.550 - 1.58i)T + (-0.786 + 0.618i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.264 + 1.83i)T + (-0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 53 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 59 | \( 1 + (0.928 + 0.371i)T^{2} \) |
| 61 | \( 1 + (0.261 - 0.273i)T + (-0.0475 - 0.998i)T^{2} \) |
| 67 | \( 1 + (0.627 + 1.81i)T + (-0.786 + 0.618i)T^{2} \) |
| 71 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (1.28 - 0.663i)T + (0.580 - 0.814i)T^{2} \) |
| 79 | \( 1 + (-0.981 + 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (0.0475 + 0.998i)T^{2} \) |
| 97 | \( 1 + (1.39 - 1.09i)T + (0.235 - 0.971i)T^{2} \) |
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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
−9.818076905430799683500677442163, −9.111387418037217285724651858780, −8.404957597007536444968797799365, −7.51337459931896011795188709998, −6.88204449893027604520839499479, −5.90981098590562872291326665400, −4.98029799337640712767059727072, −3.29517158522341122944347324444, −2.80056082048587329863407250670, −2.02671623177416754082186617015,
1.60120302223673867464325490395, 2.61871643289529769231925013733, 3.81956104097195393153025089002, 4.33897895322789298454856985428, 5.91410406770594953832993457911, 6.85442939241744388360690429292, 7.46091944577196846615195080922, 7.966996120154299173177948724414, 9.460983455542511048564872064118, 9.835698636464433586152746343533