| L(s) = 1 | + (−1 − i)5-s + 9-s + i·13-s − 2i·17-s + i·25-s − 2i·29-s + (1 − i)37-s + (−1 + i)41-s + (−1 − i)45-s − i·49-s + (1 − i)65-s + (−1 − i)73-s + 81-s + (−2 + 2i)85-s + (1 + i)89-s + ⋯ |
| L(s) = 1 | + (−1 − i)5-s + 9-s + i·13-s − 2i·17-s + i·25-s − 2i·29-s + (1 − i)37-s + (−1 + i)41-s + (−1 − i)45-s − i·49-s + (1 − i)65-s + (−1 − i)73-s + 81-s + (−2 + 2i)85-s + (1 + i)89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9392047845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9392047845\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + (1 + i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + 2iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 2iT - T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + (1 - i)T - iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-1 - i)T + iT^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.443209324123311570506459525250, −8.621745981374991045954408374429, −7.70953982723144769501676124840, −7.23902719302082973997568305311, −6.25358471597462808416465145008, −4.91134577547504508322264037924, −4.51872712443457607324640001117, −3.68351378634853099999805986151, −2.24072947671345282529433791208, −0.795084613688709256617665855902,
1.55089733187465623481227557059, 3.06555430641955359528754255813, 3.70814329232601336453696535017, 4.57493613459585662711747673934, 5.77266566402226963463868519554, 6.68924796063920943655820143661, 7.33448658195151888617681655397, 8.033814652208775802003079464087, 8.732394179160508428706444883108, 10.00698073733697786821651286726
