| L(s) = 1 | + (−0.707 − 1.58i)3-s − 2.23i·5-s − 4.24·7-s + (−2.00 + 2.23i)9-s + (−3.53 + 1.58i)15-s + (3 + 6.70i)21-s − 9.48i·23-s − 5.00·25-s + (4.94 + 1.58i)27-s − 8.94i·29-s + 9.48i·35-s − 4.47i·41-s + 12.7·43-s + (5.00 + 4.47i)45-s + 9.48i·47-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.912i)3-s − 0.999i·5-s − 1.60·7-s + (−0.666 + 0.745i)9-s + (−0.912 + 0.408i)15-s + (0.654 + 1.46i)21-s − 1.97i·23-s − 1.00·25-s + (0.952 + 0.304i)27-s − 1.66i·29-s + 1.60i·35-s − 0.698i·41-s + 1.94·43-s + (0.745 + 0.666i)45-s + 1.38i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.133376 - 0.624942i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133376 - 0.624942i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 1.58i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 9.48iT - 23T^{2} \) |
| 29 | \( 1 + 8.94iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 9.48iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 9.48iT - 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| 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
−12.17364115039456872930094446837, −10.82804725876035594364390919149, −9.697860553828572234477468062393, −8.766092299948606886619060035612, −7.69509883303102088749725745509, −6.48589477155513661571079341489, −5.81924034855598533822454440195, −4.33195339279635486193965981887, −2.55873991987141084249143622113, −0.52268030501985972374441723664,
3.04161388429217758519522527978, 3.77025411668760495326681732912, 5.49883589410066484416491785194, 6.39920909926275457995919603208, 7.33515196440029765902914468121, 9.085924980307799539241083697314, 9.770053059269143864103826346517, 10.53632331909272887233558415503, 11.39441915962966927951661169798, 12.41883650815690816896739777427
