| L(s) = 1 | − 12.4i·5-s + 12.1·7-s − 21.6i·11-s + 15.5i·13-s − 64.8·17-s + 49i·19-s − 62.4·23-s − 31·25-s + 24.9i·29-s + 24.2·31-s − 151. i·35-s + 102. i·37-s − 346.·41-s + 260i·43-s − 362.·47-s + ⋯ |
| L(s) = 1 | − 1.11i·5-s + 0.654·7-s − 0.592i·11-s + 0.332i·13-s − 0.925·17-s + 0.591i·19-s − 0.566·23-s − 0.247·25-s + 0.159i·29-s + 0.140·31-s − 0.731i·35-s + 0.454i·37-s − 1.31·41-s + 0.922i·43-s − 1.12·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6291334591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6291334591\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 12.4iT - 125T^{2} \) |
| 7 | \( 1 - 12.1T + 343T^{2} \) |
| 11 | \( 1 + 21.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 15.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 64.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 62.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 24.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 24.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 102. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 346.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 260iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 362.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 574. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 324. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 174. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 241iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 249.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 353T + 3.89e5T^{2} \) |
| 79 | \( 1 + 5.19T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 800.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
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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
−9.044655489562390410729112395024, −8.385082624458519035776026217970, −7.914369162043555924683680334155, −6.72551208176316639187324767661, −5.91407187960701020982625248195, −4.93348383457226399096978867957, −4.45631482105335292022900057943, −3.33816972752876364609978499332, −1.96449233058026634101435375619, −1.13299523070648178100570457943,
0.13340642119779418075542011671, 1.76971503066867916770339443075, 2.58014942247996528707237590430, 3.60916282668754179654662168603, 4.60463076316364490909601308795, 5.42424101147286784669384783151, 6.61774473028406286581689700363, 6.95690405495153469463453649157, 7.935106379909745103793659134686, 8.610162513030440380575827230804
