| L(s) = 1 | + 2-s − 3·3-s − 7·4-s − 5·5-s − 3·6-s − 9·7-s − 15·8-s − 18·9-s − 5·10-s + 11·11-s + 21·12-s + 2·13-s − 9·14-s + 15·15-s + 41·16-s + 21·17-s − 18·18-s − 85·19-s + 35·20-s + 27·21-s + 11·22-s + 22·23-s + 45·24-s + 25·25-s + 2·26-s + 135·27-s + 63·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.447·5-s − 0.204·6-s − 0.485·7-s − 0.662·8-s − 2/3·9-s − 0.158·10-s + 0.301·11-s + 0.505·12-s + 0.0426·13-s − 0.171·14-s + 0.258·15-s + 0.640·16-s + 0.299·17-s − 0.235·18-s − 1.02·19-s + 0.391·20-s + 0.280·21-s + 0.106·22-s + 0.199·23-s + 0.382·24-s + 1/5·25-s + 0.0150·26-s + 0.962·27-s + 0.425·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 + 9 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 + 85 T + p^{3} T^{2} \) |
| 23 | \( 1 - 22 T + p^{3} T^{2} \) |
| 29 | \( 1 + 165 T + p^{3} T^{2} \) |
| 31 | \( 1 + 83 T + p^{3} T^{2} \) |
| 37 | \( 1 - T + p^{3} T^{2} \) |
| 41 | \( 1 + 478 T + p^{3} T^{2} \) |
| 43 | \( 1 + 8 T + p^{3} T^{2} \) |
| 47 | \( 1 - 126 T + p^{3} T^{2} \) |
| 53 | \( 1 + 683 T + p^{3} T^{2} \) |
| 59 | \( 1 + 290 T + p^{3} T^{2} \) |
| 61 | \( 1 - 257 T + p^{3} T^{2} \) |
| 67 | \( 1 - 776 T + p^{3} T^{2} \) |
| 71 | \( 1 + 313 T + p^{3} T^{2} \) |
| 73 | \( 1 - 902 T + p^{3} T^{2} \) |
| 79 | \( 1 - 830 T + p^{3} T^{2} \) |
| 83 | \( 1 - 842 T + p^{3} T^{2} \) |
| 89 | \( 1 - 25 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1784 T + p^{3} T^{2} \) |
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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
−14.23818884762054503300538549627, −13.01744979915157662584740759019, −12.11542215936679308087649861172, −10.91863500584103319510480633270, −9.440842169835386193433429604798, −8.273305932618668249050131927195, −6.40740806035650295877386950202, −5.10483814859930842390914923141, −3.57122137945214328716009661090, 0,
3.57122137945214328716009661090, 5.10483814859930842390914923141, 6.40740806035650295877386950202, 8.273305932618668249050131927195, 9.440842169835386193433429604798, 10.91863500584103319510480633270, 12.11542215936679308087649861172, 13.01744979915157662584740759019, 14.23818884762054503300538549627
