L(s) = 1 | + (−2.18 + 4.71i)3-s − 14.5i·5-s − 7i·7-s + (−17.4 − 20.6i)9-s + 31.0·11-s − 55.7·13-s + (68.7 + 31.8i)15-s + 86.4i·17-s + 96.8i·19-s + (32.9 + 15.3i)21-s − 115.·23-s − 87.7·25-s + (135. − 37.0i)27-s + 49.1i·29-s − 12.5i·31-s + ⋯ |
L(s) = 1 | + (−0.420 + 0.907i)3-s − 1.30i·5-s − 0.377i·7-s + (−0.645 − 0.763i)9-s + 0.850·11-s − 1.18·13-s + (1.18 + 0.549i)15-s + 1.23i·17-s + 1.16i·19-s + (0.342 + 0.159i)21-s − 1.04·23-s − 0.701·25-s + (0.964 − 0.264i)27-s + 0.314i·29-s − 0.0728i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3590024833\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3590024833\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.18 - 4.71i)T \) |
| 7 | \( 1 + 7iT \) |
good | 5 | \( 1 + 14.5iT - 125T^{2} \) |
| 11 | \( 1 - 31.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 86.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 96.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 12.5iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 213. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 165. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 426.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 460. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 686.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 583.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 589. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 766.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 904.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 459. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 119. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 331.T + 9.12e5T^{2} \) |
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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
−11.70136759238883282744698777139, −10.34136011476257316789157574348, −9.799233938111713065340995012145, −8.827281904706021419465158713550, −8.010289552057759633360722394446, −6.46685691392779426109626789950, −5.42812009220802401719800742247, −4.49449899100874662068413442762, −3.69017829664116082319157689658, −1.49141872216298845711340809048,
0.13300623891011435106979809880, 2.10015595523158780743665878514, 3.04101459571552117070506070340, 4.85024385958063464793572545478, 6.06435328210469012427010420071, 7.00833231664581426321766845508, 7.39366687817227197513467883586, 8.824363586717863417593588683699, 9.914140905154905997314861710942, 10.92100434268557769368271653745