| L(s) = 1 | + (2.42 − 2.42i)7-s + 3i·9-s + 6.50·11-s + (−5.57 + 5.57i)17-s + 4.35i·19-s + (2.35 + 2.35i)23-s + (−3.07 − 3.07i)43-s + (8.08 − 8.08i)47-s − 4.79i·49-s − 10.8·61-s + (7.28 + 7.28i)63-s + (10.9 + 10.9i)73-s + (15.8 − 15.8i)77-s − 9·81-s + (12.3 + 12.3i)83-s + ⋯ |
| L(s) = 1 | + (0.917 − 0.917i)7-s + i·9-s + 1.96·11-s + (−1.35 + 1.35i)17-s + 0.999i·19-s + (0.491 + 0.491i)23-s + (−0.469 − 0.469i)43-s + (1.17 − 1.17i)47-s − 0.684i·49-s − 1.38·61-s + (0.917 + 0.917i)63-s + (1.28 + 1.28i)73-s + (1.80 − 1.80i)77-s − 81-s + (1.35 + 1.35i)83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.100961191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.100961191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 - 3iT^{2} \) |
| 7 | \( 1 + (-2.42 + 2.42i)T - 7iT^{2} \) |
| 11 | \( 1 - 6.50T + 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (5.57 - 5.57i)T - 17iT^{2} \) |
| 23 | \( 1 + (-2.35 - 2.35i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (3.07 + 3.07i)T + 43iT^{2} \) |
| 47 | \( 1 + (-8.08 + 8.08i)T - 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-10.9 - 10.9i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-12.3 - 12.3i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
| 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
−9.134428951880089267451656314596, −8.488924798251338631616409791315, −7.75475117003731984595173038646, −6.93527388871690230365046658313, −6.22146873800771994479017691725, −5.13306450603102277259460596513, −4.16517837059945591499380607620, −3.80075398983523786899213163507, −2.00431861782961585642768019406, −1.32708508414883122733176321214,
0.880782980934110631123568970237, 2.11799880817646622835600934798, 3.18277564359378044545203970535, 4.37148198464438839678251043111, 4.89922327863056527057525987374, 6.17914990403150385293300069867, 6.63616492166092930071615224451, 7.47735871019112768319184228902, 8.845248516444065815944748610430, 8.967946151604277408083312455910
