L(s) = 1 | − 3-s − 2.44i·5-s + (1 + 2.44i)7-s + 9-s − 2.44i·11-s − 4.89i·13-s + 2.44i·15-s − 2.44i·17-s + 2·19-s + (−1 − 2.44i)21-s − 7.34i·23-s − 0.999·25-s − 27-s + 6·29-s − 8·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.09i·5-s + (0.377 + 0.925i)7-s + 0.333·9-s − 0.738i·11-s − 1.35i·13-s + 0.632i·15-s − 0.594i·17-s + 0.458·19-s + (−0.218 − 0.534i)21-s − 1.53i·23-s − 0.199·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.899656 - 0.604456i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.899656 - 0.604456i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 5 | \( 1 + 2.44iT - 5T^{2} \) |
| 11 | \( 1 + 2.44iT - 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 2.44iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 7.34iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 7.34iT - 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 12.2iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 - 9.79iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 12.2iT - 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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
−11.42900676595020414394117403679, −10.61025254847039410134156765764, −9.400189015477704076283677412890, −8.570851329423985350218513244390, −7.80419845185953253388044449873, −6.21941121223576697317214388467, −5.38151129079929672099692054599, −4.64967213533562845333817516416, −2.84614292519676082315911144595, −0.887679951099263892192804241501,
1.78682654026469660513008450900, 3.61496018249417246476209725843, 4.63218268790506907060841087799, 6.01313725892215170231997687741, 7.13349218788953023926058641669, 7.42745352785945750077836337664, 9.120017017128087292319817463387, 10.13520756775729071601977800328, 10.84498269315449948070951377851, 11.52665316298210169516573539333