| L(s) = 1 | + 1.61i·2-s − i·3-s − 0.618·4-s + 1.61·6-s + 1.61i·7-s + 2.23i·8-s − 9-s + 4.23·11-s + 0.618i·12-s − 1.76i·13-s − 2.61·14-s − 4.85·16-s + 4.38i·17-s − 1.61i·18-s + 5·19-s + ⋯ |
| L(s) = 1 | + 1.14i·2-s − 0.577i·3-s − 0.309·4-s + 0.660·6-s + 0.611i·7-s + 0.790i·8-s − 0.333·9-s + 1.27·11-s + 0.178i·12-s − 0.489i·13-s − 0.699·14-s − 1.21·16-s + 1.06i·17-s − 0.381i·18-s + 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06286 + 1.06286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06286 + 1.06286i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 1.61iT - 2T^{2} \) |
| 7 | \( 1 - 1.61iT - 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 1.76iT - 13T^{2} \) |
| 17 | \( 1 - 4.38iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 5.47iT - 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 + 4.23iT - 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 + 11.5iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 7.09iT - 53T^{2} \) |
| 59 | \( 1 + 6.38T + 59T^{2} \) |
| 61 | \( 1 + 4.38T + 61T^{2} \) |
| 67 | \( 1 + 15.4iT - 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 1.85iT - 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 11.0iT - 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 - 18.8iT - 97T^{2} \) |
| 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
−11.80149552851293978163118270504, −10.81735852595871411471220009895, −9.365712692422313846887905581357, −8.607914522569659882159410844388, −7.68394966475850570034973880771, −6.87151211925173032238374700070, −5.97865281578356988917288761031, −5.25939197252322164535280284493, −3.51016691864182856219495816793, −1.80093876521611263405224853247,
1.17690546824124345686878891349, 2.85182433298854626272843267373, 3.87936902103617573929315218800, 4.77592678631688177209132088740, 6.44026213181268424149952149799, 7.27679093969916188874392594941, 8.851076433980686037881939114530, 9.596170133087457698792660878157, 10.28046070163483556299141251085, 11.27848243163360026424936961592
