| L(s) = 1 | − 5.19i·3-s + (10 + 15.5i)7-s − 27·9-s − 62.3i·13-s − 155. i·19-s + (81 − 51.9i)21-s − 125·25-s + 140. i·27-s − 155. i·31-s − 110·37-s − 324·39-s − 520·43-s + (−143 + 311. i)49-s − 810·57-s − 935. i·61-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + (0.539 + 0.841i)7-s − 9-s − 1.33i·13-s − 1.88i·19-s + (0.841 − 0.539i)21-s − 25-s + 1.00i·27-s − 0.903i·31-s − 0.488·37-s − 1.33·39-s − 1.84·43-s + (−0.416 + 0.908i)49-s − 1.88·57-s − 1.96i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.539i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.207626642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207626642\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| 7 | \( 1 + (-10 - 15.5i)T \) |
| good | 5 | \( 1 + 125T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 62.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 155. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 + 155. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 110T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 520T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 935. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 880T + 3.00e5T^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 - 374. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 884T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.37e3iT - 9.12e5T^{2} \) |
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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
−11.10676740560572211387858201764, −9.715851840093707844110025836994, −8.597406463243505566562423726533, −7.970873540283438168329898285891, −6.93355398191811477708425240789, −5.83022410587415024209310344723, −4.99616537540184709874014994255, −3.07939487069759577264637727264, −2.01585645639853030158306634718, −0.41721417423514636467542934866,
1.71352074137140589484897419982, 3.58956220414985624183442629108, 4.31729940070952066850243266248, 5.41973963683204377947770294299, 6.63992962171986762516218060969, 7.87925110755520057981937380345, 8.753491919803755979967730197630, 9.875874426186003541709374045325, 10.40789804648586078280810315332, 11.45382503926397097422445012845
