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
−10.89139937219884820317336108875, −9.452576728197947626447469686149, −8.435165563425194873724536702224, −8.024524677016151818786350404799, −6.92689809232153319449887134998, −6.01279344882289101371031370157, −4.71450084485900981237182951538, −3.99373598584491234803236427291, −2.17793362240987473037361193923, −0.75510460038639689878501717743,
1.80925837671299582998309929030, 3.56388167647729392741235185502, 4.37830292412384870489032868945, 5.16607881861660906894003778188, 6.74037492801591225859611996898, 7.35103323894061679605493327585, 8.397120046836998017712200695869, 9.267064188176406411171777126443, 10.40160987991784136842067616782, 11.16697170136265553170053254573