L(s) = 1 | + 2-s + (1.68 − 0.420i)3-s + 4-s + (−1.08 + 1.95i)5-s + (1.68 − 0.420i)6-s + (−0.595 − 2.57i)7-s + 8-s + (2.64 − 1.41i)9-s + (−1.08 + 1.95i)10-s + 2.82i·11-s + (1.68 − 0.420i)12-s − 3.36·13-s + (−0.595 − 2.57i)14-s + (−1 + 3.74i)15-s + 16-s − 4.75i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.970 − 0.242i)3-s + 0.5·4-s + (−0.485 + 0.874i)5-s + (0.685 − 0.171i)6-s + (−0.224 − 0.974i)7-s + 0.353·8-s + (0.881 − 0.471i)9-s + (−0.343 + 0.618i)10-s + 0.852i·11-s + (0.485 − 0.121i)12-s − 0.931·13-s + (−0.159 − 0.688i)14-s + (−0.258 + 0.966i)15-s + 0.250·16-s − 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.10970 - 0.0361380i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.10970 - 0.0361380i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.68 + 0.420i)T \) |
| 5 | \( 1 + (1.08 - 1.95i)T \) |
| 7 | \( 1 + (0.595 + 2.57i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 4.75iT - 17T^{2} \) |
| 19 | \( 1 - 5.59iT - 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 + 0.500iT - 29T^{2} \) |
| 31 | \( 1 + 3.06iT - 31T^{2} \) |
| 37 | \( 1 - 3.32iT - 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 7.82iT - 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 + 2.16T + 59T^{2} \) |
| 61 | \( 1 + 2.52iT - 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 9.81iT - 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 - 3.29T + 79T^{2} \) |
| 83 | \( 1 - 6.97iT - 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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
−12.43583737997147497946817679951, −11.68442962472058292083243227644, −10.21488383632231225172212468712, −9.776739549078557406913921056693, −7.82837325931651539223437472339, −7.41953217664834026855004514536, −6.43659011311274314839627121834, −4.48207521790076744158141650036, −3.57594688228586469773057112953, −2.31047984875422437505398978778,
2.28212128606342606536019628260, 3.63020504440755169683980598329, 4.75281822130228883665904586598, 5.90619251711789492644004616728, 7.47353212963854420461898893611, 8.506637451408448899741346540402, 9.156121062782976811693835582394, 10.42853972531136794645899788321, 11.75636660651483885119308070871, 12.53060932893599816798698930245