L(s) = 1 | − 5.83i·5-s − 8.83i·7-s − 23.6·11-s − 54.6·13-s + 117. i·17-s − 109. i·19-s − 33.5·23-s + 90.9·25-s + 40.0i·29-s + 292. i·31-s − 51.5·35-s − 283.·37-s − 367. i·41-s + 323. i·43-s + 66.2·47-s + ⋯ |
L(s) = 1 | − 0.522i·5-s − 0.476i·7-s − 0.647·11-s − 1.16·13-s + 1.67i·17-s − 1.32i·19-s − 0.304·23-s + 0.727·25-s + 0.256i·29-s + 1.69i·31-s − 0.249·35-s − 1.25·37-s − 1.39i·41-s + 1.14i·43-s + 0.205·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.517245914\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.517245914\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.83iT - 125T^{2} \) |
| 7 | \( 1 + 8.83iT - 343T^{2} \) |
| 11 | \( 1 + 23.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 109. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 33.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 40.0iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 292. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 367. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 323. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 66.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 158. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 848.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 348.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 194. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 939.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 473.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 273. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 338.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 739. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 448.T + 9.12e5T^{2} \) |
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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
−8.774764993507113655847494299656, −8.350989785938276676329298377451, −7.24751111856124415694003703535, −6.78991139528305802138125133276, −5.50625536914516075489903625973, −4.91737395594279879403977319724, −4.03878303303775294221280147378, −2.93173053243105515230139541697, −1.85323864361747824132731149838, −0.63539850038735781088529678807,
0.50516639902771721452694085964, 2.19467459758766865814649773854, 2.74918609990779434302988359839, 3.88116467000379093505281879990, 5.04785763783826690251876873694, 5.57009103870119087107549345878, 6.67991114119424003298592751431, 7.39006824756841688084144413857, 8.047721780210572364391018884307, 9.012985270046454003926910183491