| L(s) = 1 | + 3-s + 3i·5-s + 3i·7-s − 2·9-s + (2 − 3i)13-s + 3i·15-s + 3·17-s − 6i·19-s + 3i·21-s + 6·23-s − 4·25-s − 5·27-s − 9·35-s − 3i·37-s + (2 − 3i)39-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34i·5-s + 1.13i·7-s − 0.666·9-s + (0.554 − 0.832i)13-s + 0.774i·15-s + 0.727·17-s − 1.37i·19-s + 0.654i·21-s + 1.25·23-s − 0.800·25-s − 0.962·27-s − 1.52·35-s − 0.493i·37-s + (0.320 − 0.480i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21299 + 0.649175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21299 + 0.649175i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-2 + 3i)T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 15iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−12.56591945348220648660290116662, −11.34913759877931000882715344370, −10.79841486861687876145581655720, −9.455261620798328003273195771539, −8.621273572444831610393501303142, −7.55056661700761708834136011260, −6.36392121518998361378249381242, −5.33610604815166923439313883019, −3.22516482017092836675604231153, −2.64144545797811667872857088370,
1.32750074296120482138943209608, 3.48628648044729452992003112336, 4.59575373214872018470438913342, 5.88295087585444615924522111830, 7.39812493034566573189772641080, 8.382982560575028275442200366894, 9.085027294343770777167117739192, 10.15131982791510938619243267869, 11.34321801490786955910351958186, 12.34696847050935546684804652172
