| L(s) = 1 | − 3-s + (−1 − 2i)5-s − 4i·7-s + 9-s + 6·13-s + (1 + 2i)15-s − 2i·17-s + 6i·19-s + 4i·21-s + 6i·23-s + (−3 + 4i)25-s − 27-s + 8i·29-s + 8·31-s + (−8 + 4i)35-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.447 − 0.894i)5-s − 1.51i·7-s + 0.333·9-s + 1.66·13-s + (0.258 + 0.516i)15-s − 0.485i·17-s + 1.37i·19-s + 0.872i·21-s + 1.25i·23-s + (−0.600 + 0.800i)25-s − 0.192·27-s + 1.48i·29-s + 1.43·31-s + (−1.35 + 0.676i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.542600077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542600077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - 14iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.379916880006925162472316143505, −7.61968180557783570759782479667, −7.16181638274302963968502988478, −6.05094770933063583748231132730, −5.59906536980776722814129760033, −4.36481820466417153689628669151, −4.12793775552164044992206194290, −3.23268003449758948079940479413, −1.24434322889413472729196767475, −1.04304104692219698330732547540,
0.66948686774288900859461338694, 2.28126983532746710351164822964, 2.84615067839904685560346464794, 3.99136953022986276516737174272, 4.69836325677613360223673500981, 5.91296591980457636011296854570, 6.17265951842704904426305040311, 6.78110007517117028618950061742, 7.967314124798459046028881949536, 8.411772705319067576721238517919
