L(s) = 1 | + (−0.565 − 0.978i)5-s + (3.71 + 2.14i)7-s + (1.00 + 0.582i)11-s + (−2.64 + 1.52i)13-s − 1.49i·17-s + 3.42·19-s + (3.85 + 6.68i)23-s + (1.86 − 3.22i)25-s + (−0.709 + 1.22i)29-s + (−4.66 + 2.69i)31-s − 4.84i·35-s + 2.97i·37-s + (4.23 − 2.44i)41-s + (1.74 − 3.01i)43-s + (1.77 − 3.08i)47-s + ⋯ |
L(s) = 1 | + (−0.252 − 0.437i)5-s + (1.40 + 0.810i)7-s + (0.304 + 0.175i)11-s + (−0.733 + 0.423i)13-s − 0.362i·17-s + 0.785·19-s + (0.804 + 1.39i)23-s + (0.372 − 0.644i)25-s + (−0.131 + 0.228i)29-s + (−0.837 + 0.483i)31-s − 0.819i·35-s + 0.488i·37-s + (0.661 − 0.381i)41-s + (0.265 − 0.460i)43-s + (0.259 − 0.449i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71344 + 0.341515i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71344 + 0.341515i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.565 + 0.978i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.71 - 2.14i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.00 - 0.582i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.64 - 1.52i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.49iT - 17T^{2} \) |
| 19 | \( 1 - 3.42T + 19T^{2} \) |
| 23 | \( 1 + (-3.85 - 6.68i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.709 - 1.22i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.66 - 2.69i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.97iT - 37T^{2} \) |
| 41 | \( 1 + (-4.23 + 2.44i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.74 + 3.01i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.77 + 3.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + (-7.50 + 4.33i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.16 - 1.82i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.58 - 9.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 + (2.24 + 1.29i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.98 + 2.30i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 8.63iT - 89T^{2} \) |
| 97 | \( 1 + (-3.35 + 5.81i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.17229884663821231291872845580, −9.134823373807428449557686508450, −8.689303718757697052297076274534, −7.64825894282309067119870024503, −7.02423252442600789763919672170, −5.44865527168275934564261036147, −5.09191459117847771745037004334, −4.00600062457735310815078931118, −2.52773216895525152338821182476, −1.35155170341768193828353345207,
1.03372211774940829856110216049, 2.51388066997176797908778747473, 3.82013535614669279580939520417, 4.71161879434478714082886008604, 5.60514796148474723195962804986, 6.95550113069375539671776847783, 7.51420614657707658661109179090, 8.274418621670987048085013805312, 9.251759981543514093850183706449, 10.32406515951758814121377442240