L(s) = 1 | + (−0.658 + 1.25i)2-s + (−1.13 − 1.64i)4-s + 2.08i·5-s + (2.80 − 0.333i)8-s + (−2.60 − 1.37i)10-s + 4.26·11-s − 4.80·13-s + (−1.43 + 3.73i)16-s − 3.20i·17-s + 2.81i·19-s + (3.42 − 2.35i)20-s + (−2.80 + 5.34i)22-s − 4.66·23-s + 0.669·25-s + (3.16 − 6.01i)26-s + ⋯ |
L(s) = 1 | + (−0.465 + 0.885i)2-s + (−0.566 − 0.823i)4-s + 0.930i·5-s + (0.993 − 0.117i)8-s + (−0.823 − 0.433i)10-s + 1.28·11-s − 1.33·13-s + (−0.357 + 0.933i)16-s − 0.777i·17-s + 0.646i·19-s + (0.766 − 0.527i)20-s + (−0.599 + 1.13i)22-s − 0.971·23-s + 0.133·25-s + (0.620 − 1.17i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8248001990\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8248001990\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.658 - 1.25i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.08iT - 5T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 17 | \( 1 + 3.20iT - 17T^{2} \) |
| 19 | \( 1 - 2.81iT - 19T^{2} \) |
| 23 | \( 1 + 4.66T + 23T^{2} \) |
| 29 | \( 1 - 3.87iT - 29T^{2} \) |
| 31 | \( 1 - 10.2iT - 31T^{2} \) |
| 37 | \( 1 - 0.273T + 37T^{2} \) |
| 41 | \( 1 - 0.387iT - 41T^{2} \) |
| 43 | \( 1 + 0.907iT - 43T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 - 11.7iT - 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 + 1.40iT - 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 0.826iT - 79T^{2} \) |
| 83 | \( 1 + 5.69T + 83T^{2} \) |
| 89 | \( 1 - 3.02iT - 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.498757965638129317612763120637, −8.988134252857954534694594474860, −7.954217627117337895217068763856, −7.12173723932351975684136710955, −6.79341087963735453490542518092, −5.88354656678433720871187287842, −4.94471707865189877094638080674, −4.01071523349103255822683602850, −2.80430423583573295381743413524, −1.42852357792577046662499520543,
0.37979472306125781040975661393, 1.62233611148280031140669214392, 2.58563089157294418640958667379, 4.00192830892392899161203344113, 4.39665036350186931798414229498, 5.47022035025323366878978222479, 6.61251181195301188908094143515, 7.63302800258705511995831004720, 8.291089660585550620663823989700, 9.156666312314604098391750573544