L(s) = 1 | + (−0.304 + 1.13i)3-s + (−1.79 + 1.33i)5-s + (2.55 − 0.698i)7-s + (1.40 + 0.810i)9-s + (0.371 + 0.643i)11-s + (2.05 + 2.05i)13-s + (−0.975 − 2.43i)15-s + (−6.33 − 1.69i)17-s + (−0.946 + 1.63i)19-s + (0.0172 + 3.10i)21-s + (1.36 + 5.11i)23-s + (1.41 − 4.79i)25-s + (−3.83 + 3.83i)27-s + 9.69i·29-s + (−2.96 + 1.71i)31-s + ⋯ |
L(s) = 1 | + (−0.175 + 0.655i)3-s + (−0.800 + 0.599i)5-s + (0.964 − 0.264i)7-s + (0.467 + 0.270i)9-s + (0.112 + 0.194i)11-s + (0.570 + 0.570i)13-s + (−0.251 − 0.629i)15-s + (−1.53 − 0.411i)17-s + (−0.217 + 0.375i)19-s + (0.00376 + 0.678i)21-s + (0.285 + 1.06i)23-s + (0.282 − 0.959i)25-s + (−0.738 + 0.738i)27-s + 1.79i·29-s + (−0.532 + 0.307i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.729219 + 0.977931i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.729219 + 0.977931i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.79 - 1.33i)T \) |
| 7 | \( 1 + (-2.55 + 0.698i)T \) |
good | 3 | \( 1 + (0.304 - 1.13i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-0.371 - 0.643i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.05 - 2.05i)T + 13iT^{2} \) |
| 17 | \( 1 + (6.33 + 1.69i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.946 - 1.63i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.36 - 5.11i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 9.69iT - 29T^{2} \) |
| 31 | \( 1 + (2.96 - 1.71i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.58 + 0.691i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 0.817iT - 41T^{2} \) |
| 43 | \( 1 + (1.59 - 1.59i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.21 + 4.54i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (4.81 + 1.29i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.27 - 2.20i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.25 - 3.03i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.54 + 13.2i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + (2.29 - 8.54i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.70 - 3.29i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.23 + 9.23i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.01 + 5.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.16 + 3.16i)T - 97iT^{2} \) |
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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
−11.16697170136265553170053254573, −10.40160987991784136842067616782, −9.267064188176406411171777126443, −8.397120046836998017712200695869, −7.35103323894061679605493327585, −6.74037492801591225859611996898, −5.16607881861660906894003778188, −4.37830292412384870489032868945, −3.56388167647729392741235185502, −1.80925837671299582998309929030,
0.75510460038639689878501717743, 2.17793362240987473037361193923, 3.99373598584491234803236427291, 4.71450084485900981237182951538, 6.01279344882289101371031370157, 6.92689809232153319449887134998, 8.024524677016151818786350404799, 8.435165563425194873724536702224, 9.452576728197947626447469686149, 10.89139937219884820317336108875