L(s) = 1 | − i·2-s + (1.59 + 0.684i)3-s − 4-s + 4.42·5-s + (0.684 − 1.59i)6-s + (−2.43 − 1.02i)7-s + i·8-s + (2.06 + 2.17i)9-s − 4.42i·10-s − 5.78i·11-s + (−1.59 − 0.684i)12-s + i·13-s + (−1.02 + 2.43i)14-s + (7.03 + 3.02i)15-s + 16-s − 2.97·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.918 + 0.395i)3-s − 0.5·4-s + 1.97·5-s + (0.279 − 0.649i)6-s + (−0.921 − 0.387i)7-s + 0.353i·8-s + (0.687 + 0.726i)9-s − 1.39i·10-s − 1.74i·11-s + (−0.459 − 0.197i)12-s + 0.277i·13-s + (−0.274 + 0.651i)14-s + (1.81 + 0.781i)15-s + 0.250·16-s − 0.721·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12601 - 0.904422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12601 - 0.904422i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-1.59 - 0.684i)T \) |
| 7 | \( 1 + (2.43 + 1.02i)T \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 - 4.42T + 5T^{2} \) |
| 11 | \( 1 + 5.78iT - 11T^{2} \) |
| 17 | \( 1 + 2.97T + 17T^{2} \) |
| 19 | \( 1 + 2.66iT - 19T^{2} \) |
| 23 | \( 1 - 6.57iT - 23T^{2} \) |
| 29 | \( 1 - 3.44iT - 29T^{2} \) |
| 31 | \( 1 - 3.06iT - 31T^{2} \) |
| 37 | \( 1 + 4.71T + 37T^{2} \) |
| 41 | \( 1 - 2.07T + 41T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 + 4.28T + 47T^{2} \) |
| 53 | \( 1 - 5.36iT - 53T^{2} \) |
| 59 | \( 1 - 2.04T + 59T^{2} \) |
| 61 | \( 1 + 7.83iT - 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 + 4.46iT - 71T^{2} \) |
| 73 | \( 1 + 4.11iT - 73T^{2} \) |
| 79 | \( 1 - 1.88T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 5.79iT - 97T^{2} \) |
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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
−10.50933251140227790779001326521, −9.752795346886929619074694754015, −9.129640923207919336463488073144, −8.613259233297300472680466249889, −6.95997848072249449313324933566, −5.98399570800883309523382055604, −5.00601673853349970075356113560, −3.46935100501053697695460981992, −2.78755445690264734795008357779, −1.52826153776323574319309850129,
1.87641972468384453349681771771, 2.70083902407770049367680535672, 4.39812481825891421823072431489, 5.62453313658762661373702948827, 6.57433953652247767970585354613, 6.98015055492624540903580428883, 8.392466548465372393969924171758, 9.184244023406120164743946881250, 9.906340281657679146810429468273, 10.16863153424200676026775588507