| L(s) = 1 | + 6.10·2-s + (−8.98 + 12.7i)3-s + 5.21·4-s + 25i·5-s + (−54.8 + 77.7i)6-s + 135. i·7-s − 163.·8-s + (−81.5 − 228. i)9-s + 152. i·10-s + (−8.52 + 401. i)11-s + (−46.8 + 66.4i)12-s − 924. i·13-s + 823. i·14-s + (−318. − 224. i)15-s − 1.16e3·16-s + 526.·17-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + (−0.576 + 0.817i)3-s + 0.162·4-s + 0.447i·5-s + (−0.621 + 0.881i)6-s + 1.04i·7-s − 0.902·8-s + (−0.335 − 0.941i)9-s + 0.482i·10-s + (−0.0212 + 0.999i)11-s + (−0.0938 + 0.133i)12-s − 1.51i·13-s + 1.12i·14-s + (−0.365 − 0.257i)15-s − 1.13·16-s + 0.441·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.593 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2364374906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2364374906\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.98 - 12.7i)T \) |
| 5 | \( 1 - 25iT \) |
| 11 | \( 1 + (8.52 - 401. i)T \) |
| good | 2 | \( 1 - 6.10T + 32T^{2} \) |
| 7 | \( 1 - 135. iT - 1.68e4T^{2} \) |
| 13 | \( 1 + 924. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 526.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 752. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 126.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.43e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.89e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.28e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.13e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.41e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.56e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.15e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.89e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 5.66e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.85e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 8.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.09e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 7.54e4T + 8.58e9T^{2} \) |
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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
−12.42408298370857069699065107108, −11.88988599646050909322734667022, −10.69925713577424820285883599537, −9.690980872821652250310154506116, −8.703644917621996205503888982010, −6.94626208227370496700224410518, −5.54888106464935349611733840254, −5.21883949499091335634917818387, −3.78359884195186468486421458332, −2.70658279354370701428362701329,
0.05718796944919661886832925864, 1.54209612918496151895603843854, 3.53974390200113721297809755144, 4.64349423168740099715758838647, 5.78344945549326724685727432233, 6.67223392495920813759211275927, 7.913618645171386070282484185660, 9.143201397101817288992764111696, 10.65211857742635945565961494465, 11.69333178663210648694677645421
