L(s) = 1 | + (0.5 − 0.866i)2-s + (1.25 + 1.19i)3-s + (−0.499 − 0.866i)4-s + (1.80 − 1.04i)5-s + (1.66 − 0.492i)6-s + (1.78 − 1.95i)7-s − 0.999·8-s + (0.160 + 2.99i)9-s − 2.08i·10-s − 2.15·11-s + (0.403 − 1.68i)12-s + (−0.217 − 3.59i)13-s + (−0.800 − 2.52i)14-s + (3.50 + 0.839i)15-s + (−0.5 + 0.866i)16-s + (−0.278 − 0.482i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.725 + 0.687i)3-s + (−0.249 − 0.433i)4-s + (0.806 − 0.465i)5-s + (0.677 − 0.201i)6-s + (0.674 − 0.738i)7-s − 0.353·8-s + (0.0534 + 0.998i)9-s − 0.658i·10-s − 0.650·11-s + (0.116 − 0.486i)12-s + (−0.0602 − 0.998i)13-s + (−0.213 − 0.673i)14-s + (0.905 + 0.216i)15-s + (−0.125 + 0.216i)16-s + (−0.0675 − 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.26658 - 0.959100i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.26658 - 0.959100i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.25 - 1.19i)T \) |
| 7 | \( 1 + (-1.78 + 1.95i)T \) |
| 13 | \( 1 + (0.217 + 3.59i)T \) |
good | 5 | \( 1 + (-1.80 + 1.04i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 17 | \( 1 + (0.278 + 0.482i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 + (-1.79 - 1.03i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.10 - 3.52i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.21 - 5.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.20 - 4.15i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.532 + 0.307i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.33 + 7.50i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.507 - 0.292i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.68 + 3.85i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.35 + 0.782i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 8.20iT - 61T^{2} \) |
| 67 | \( 1 + 14.2iT - 67T^{2} \) |
| 71 | \( 1 + (6.52 - 11.3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.198 - 0.344i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.73 - 9.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 + (7.54 + 4.35i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.64 - 2.85i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.65413622385413756805847149301, −9.871618362228856643503506567977, −9.201645741688171554586238575551, −8.165497626737187792851217972183, −7.31790798984275560275235987719, −5.41400644798760375691391535277, −5.12172159053964069459482685123, −3.84630457900342020659162276848, −2.79703532267991054750864645333, −1.46615671624392944358250563099,
1.94249192754755690922584446783, 2.81755244189659220037511186437, 4.31991370956071161341288410694, 5.65514645792779935461370055980, 6.27914389909858897922485745216, 7.42888096503940798467444969774, 7.979210166888117995836626448875, 9.157356083186412702204451377264, 9.581100931787266490845492600717, 11.10473388725836316549952148296