| L(s) = 1 | + (2 − 11i)5-s + 27i·9-s + (37 + 37i)13-s + (99 − 99i)17-s + (−117 − 44i)25-s − 284i·29-s + (91 − 91i)37-s + 472·41-s + (297 + 54i)45-s − 343i·49-s + (27 + 27i)53-s + 468·61-s + (481 − 333i)65-s + (253 + 253i)73-s − 729·81-s + ⋯ |
| L(s) = 1 | + (0.178 − 0.983i)5-s + i·9-s + (0.789 + 0.789i)13-s + (1.41 − 1.41i)17-s + (−0.936 − 0.351i)25-s − 1.81i·29-s + (0.404 − 0.404i)37-s + 1.79·41-s + (0.983 + 0.178i)45-s − i·49-s + (0.0699 + 0.0699i)53-s + 0.982·61-s + (0.917 − 0.635i)65-s + (0.405 + 0.405i)73-s − 0.999·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.984691602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.984691602\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 + 11i)T \) |
| good | 3 | \( 1 - 27iT^{2} \) |
| 7 | \( 1 + 343iT^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + (-37 - 37i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-99 + 99i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4iT^{2} \) |
| 29 | \( 1 + 284iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 + (-91 + 91i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 472T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4iT^{2} \) |
| 47 | \( 1 + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-27 - 27i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 468T + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5iT^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 + (-253 - 253i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 - 5.71e5iT^{2} \) |
| 89 | \( 1 + 176iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (611 - 611i)T - 9.12e5iT^{2} \) |
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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
−11.26734741273300891822193368054, −9.987823816311971839973884287537, −9.282214400625587334733670394429, −8.221563128706653465983801246775, −7.45376523736800120752610399821, −5.97282510828134802573881404735, −5.08739688346561267313111985571, −4.04813940799823796442206841176, −2.32425013519407610496182098717, −0.880545873266245522058726540880,
1.22570856556852482188672804371, 3.05328065203054879033441404767, 3.80975708899916515805137989711, 5.66847457086027759892950313476, 6.31856337765354420127554316574, 7.43929838839368729666204624753, 8.435944481166980822423267803535, 9.584976507066331795132341414539, 10.46502509883899556893388036471, 11.11275981506197055708958060146
