L(s) = 1 | + (−0.707 − 0.707i)5-s + 1.73i·7-s + (0.505 + 0.505i)11-s + (−1.88 + 1.88i)13-s − 4.53·17-s + (3.22 − 3.22i)19-s − 8.85i·23-s + 1.00i·25-s + (2.44 − 2.44i)29-s + 5.70·31-s + (1.22 − 1.22i)35-s + (−5.35 − 5.35i)37-s + 10.0i·41-s + (2.10 + 2.10i)43-s + 4.32·47-s + ⋯ |
L(s) = 1 | + (−0.316 − 0.316i)5-s + 0.656i·7-s + (0.152 + 0.152i)11-s + (−0.523 + 0.523i)13-s − 1.09·17-s + (0.738 − 0.738i)19-s − 1.84i·23-s + 0.200i·25-s + (0.453 − 0.453i)29-s + 1.02·31-s + (0.207 − 0.207i)35-s + (−0.880 − 0.880i)37-s + 1.56i·41-s + (0.321 + 0.321i)43-s + 0.630·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.514716596\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.514716596\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 + (-0.505 - 0.505i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.88 - 1.88i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.53T + 17T^{2} \) |
| 19 | \( 1 + (-3.22 + 3.22i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.85iT - 23T^{2} \) |
| 29 | \( 1 + (-2.44 + 2.44i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 + (5.35 + 5.35i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.0iT - 41T^{2} \) |
| 43 | \( 1 + (-2.10 - 2.10i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 + (-1.37 - 1.37i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.64 - 6.64i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.26 + 5.26i)T - 61iT^{2} \) |
| 67 | \( 1 + (-10.5 + 10.5i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.0iT - 71T^{2} \) |
| 73 | \( 1 - 6.63iT - 73T^{2} \) |
| 79 | \( 1 + 4.27T + 79T^{2} \) |
| 83 | \( 1 + (-9.15 + 9.15i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.23iT - 89T^{2} \) |
| 97 | \( 1 - 1.94T + 97T^{2} \) |
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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
−8.682848627236624632502758475345, −8.176589174385742989000653576411, −7.04803450579820980283693072513, −6.60667447214444559136946598717, −5.60412031288861845154145224977, −4.66126576094150137691779392609, −4.24774181202237598059680256246, −2.83294359107631061236270559824, −2.18851946516140603382024544798, −0.64946134826685848737088672857,
0.872608034208744197971800362498, 2.18191738594498720835375407124, 3.34390822318326289725520575670, 3.91789091229275338888275798879, 4.97906023408319339544955797226, 5.69541584328655985025156466888, 6.77169730309990484903209178584, 7.26764851558440355198492269479, 7.980935102345099222621907061640, 8.770859334900572856678402049482