| L(s) = 1 | + (7 + 7i)7-s − 10·11-s + (−9 + 9i)13-s + (1 + i)17-s − 8i·19-s + (−23 + 23i)23-s + 8i·29-s − 14·31-s + (−33 − 33i)37-s + 14·41-s + (15 − 15i)43-s + (−39 − 39i)47-s + 49i·49-s + (−7 + 7i)53-s + 56i·59-s + ⋯ |
| L(s) = 1 | + (1 + i)7-s − 0.909·11-s + (−0.692 + 0.692i)13-s + (0.0588 + 0.0588i)17-s − 0.421i·19-s + (−1 + i)23-s + 0.275i·29-s − 0.451·31-s + (−0.891 − 0.891i)37-s + 0.341·41-s + (0.348 − 0.348i)43-s + (−0.829 − 0.829i)47-s + 0.999i·49-s + (−0.132 + 0.132i)53-s + 0.949i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8943377885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8943377885\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-7 - 7i)T + 49iT^{2} \) |
| 11 | \( 1 + 10T + 121T^{2} \) |
| 13 | \( 1 + (9 - 9i)T - 169iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + 289iT^{2} \) |
| 19 | \( 1 + 8iT - 361T^{2} \) |
| 23 | \( 1 + (23 - 23i)T - 529iT^{2} \) |
| 29 | \( 1 - 8iT - 841T^{2} \) |
| 31 | \( 1 + 14T + 961T^{2} \) |
| 37 | \( 1 + (33 + 33i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 14T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-15 + 15i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (39 + 39i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (7 - 7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 56iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 42T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-7 - 7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 98T + 5.04e3T^{2} \) |
| 73 | \( 1 + (49 - 49i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 96iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (63 - 63i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 112iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (33 + 33i)T + 9.40e3iT^{2} \) |
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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
−10.25046538391802174955620389014, −9.341157447893093095193916010248, −8.578489733077114654031251866307, −7.78525085917067029954359474848, −6.98567422437784108828928275042, −5.61990739702584192083379368300, −5.18934719854384105355372680623, −4.06160569411435278487264954591, −2.60470897327465156616230835657, −1.77211272289172117505479798148,
0.27140448244139971604358331831, 1.75197741084620801753030296099, 3.02629437341443782897548160355, 4.32304652082231025872371308324, 5.00055288958139290662705393646, 6.03379653537262315605098217498, 7.25771895393379382985012563011, 7.86495731090222386467269694895, 8.462373680623894977622339467094, 9.852967374162478692001198741465
