| L(s) = 1 | + i·2-s + i·3-s + 4-s + (1 − 2i)5-s − 6-s + i·7-s + 3i·8-s − 9-s + (2 + i)10-s − 6·11-s + i·12-s − 2i·13-s − 14-s + (2 + i)15-s − 16-s − 4i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s + 0.5·4-s + (0.447 − 0.894i)5-s − 0.408·6-s + 0.377i·7-s + 1.06i·8-s − 0.333·9-s + (0.632 + 0.316i)10-s − 1.80·11-s + 0.288i·12-s − 0.554i·13-s − 0.267·14-s + (0.516 + 0.258i)15-s − 0.250·16-s − 0.970i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.987759 + 0.610468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.987759 + 0.610468i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−14.07765741036902913605419866632, −13.03878561228530477495552939210, −11.86948020238545760806007327417, −10.69192697832386041230196696032, −9.590614160303390257499736002489, −8.370119229283450604227174471651, −7.40285334522283855310331904915, −5.53092384774540902824306223888, −5.20934549967759721253207672094, −2.70173620439600678454391775595,
2.06135419027838012408223731157, 3.31926418578659151487141795523, 5.66673716415926716727854405042, 6.96295836787847011215570227142, 7.77449242194542902577294645997, 9.704083858777687059690286950373, 10.65843719583477371813672800041, 11.27433118061208772548131650751, 12.59576519015341727507158702442, 13.34777610896378407247861108327
