L(s) = 1 | + (−4.58 + 2.44i)3-s − 17.2i·5-s − 0.269i·7-s + (15 − 22.4i)9-s − 41.1·11-s − 64.6·13-s + (42.3 + 79.1i)15-s + 63.4i·17-s + 154. i·19-s + (0.660 + 1.23i)21-s − 43.3·23-s − 173.·25-s + (−13.7 + 139. i)27-s − 240. i·29-s − 79.1i·31-s + ⋯ |
L(s) = 1 | + (−0.881 + 0.471i)3-s − 1.54i·5-s − 0.0145i·7-s + (0.555 − 0.831i)9-s − 1.12·11-s − 1.37·13-s + (0.728 + 1.36i)15-s + 0.904i·17-s + 1.85i·19-s + (0.00686 + 0.0128i)21-s − 0.392·23-s − 1.38·25-s + (−0.0979 + 0.995i)27-s − 1.54i·29-s − 0.458i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7695837386\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7695837386\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.58 - 2.44i)T \) |
good | 5 | \( 1 + 17.2iT - 125T^{2} \) |
| 7 | \( 1 + 0.269iT - 343T^{2} \) |
| 11 | \( 1 + 41.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 154. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 43.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 240. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 79.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 101. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 120. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 6.17iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 45.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 651.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 685. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 836.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 285.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 940. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 432. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−9.975443497106310464758874526931, −9.393161797703098195608736956703, −8.175368446390737502409617902628, −7.65842237131498995351665388671, −6.06795136880184976853864391714, −5.49634714795935411101453107446, −4.66148138386272860676293817215, −3.92508711989903605101529482480, −2.05993351973430828783617513017, −0.66381046606646293253266333819,
0.37327814523938836551226926751, 2.35143132665713856683189033052, 2.91098525352400804507900513584, 4.70481647364890971067022813649, 5.39633138098914126744100041230, 6.57258480287318163606081642953, 7.25001959352698929030363881697, 7.58246689595047694285926130473, 9.164799633565425781421775069734, 10.25745074177489927657023269529