L(s) = 1 | + (−1.83 + 3.18i)5-s + (2.63 − 0.277i)7-s + (−0.301 − 0.522i)11-s − 5.25·13-s + (−2.12 − 3.68i)17-s + (3.68 − 6.38i)19-s + (−0.578 + 1.00i)23-s + (−4.25 − 7.37i)25-s − 7.97·29-s + (−1.57 − 2.72i)31-s + (−3.95 + 8.88i)35-s + (0.00266 − 0.00462i)37-s − 4.01·41-s + 7.32·43-s + (6.10 − 10.5i)47-s + ⋯ |
L(s) = 1 | + (−0.822 + 1.42i)5-s + (0.994 − 0.104i)7-s + (−0.0909 − 0.157i)11-s − 1.45·13-s + (−0.515 − 0.892i)17-s + (0.845 − 1.46i)19-s + (−0.120 + 0.209i)23-s + (−0.851 − 1.47i)25-s − 1.48·29-s + (−0.282 − 0.490i)31-s + (−0.668 + 1.50i)35-s + (0.000438 − 0.000760i)37-s − 0.627·41-s + 1.11·43-s + (0.891 − 1.54i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8226309676\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8226309676\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.277i)T \) |
good | 5 | \( 1 + (1.83 - 3.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.301 + 0.522i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.25T + 13T^{2} \) |
| 17 | \( 1 + (2.12 + 3.68i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.68 + 6.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.578 - 1.00i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 + (1.57 + 2.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.00266 + 0.00462i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.01T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 + (-6.10 + 10.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.64 - 8.05i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.30 - 5.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.969 + 1.67i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.31 + 7.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.13T + 71T^{2} \) |
| 73 | \( 1 + (5.33 + 9.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.07 - 3.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + (-4.09 + 7.09i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.895170236784106834889277108722, −7.66373004045774737215974247448, −7.37373655364092900054466506423, −6.92664700331957574121547994198, −5.59569627422367919314952591260, −4.81372927824692254780822308534, −3.95173451052721461634016307310, −2.87400771410989854285277761504, −2.23549534970174117098286041375, −0.29619063254712748247880875352,
1.22542791293841938259572076911, 2.17526712064444449399440035426, 3.77120439842200442358276467088, 4.39727791246667858318357166152, 5.18418845412640632749210186597, 5.71947213196051896820184083637, 7.22846641949804175154203779724, 7.78253247806258707280469935388, 8.322485939378483191399814584434, 9.070284865060273623932645894949