| L(s) = 1 | + 1.41·2-s − 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (6.70 − 2.02i)7-s + 2.82·8-s − 2.99·9-s − 5.00·11-s − 3.46i·12-s + 9.06i·13-s + (9.47 − 2.86i)14-s + 4.00·16-s − 19.7i·17-s − 4.24·18-s − 25.6i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (0.957 − 0.288i)7-s + 0.353·8-s − 0.333·9-s − 0.455·11-s − 0.288i·12-s + 0.697i·13-s + (0.676 − 0.204i)14-s + 0.250·16-s − 1.16i·17-s − 0.235·18-s − 1.35i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.256911453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.256911453\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.70 + 2.02i)T \) |
| good | 11 | \( 1 + 5.00T + 121T^{2} \) |
| 13 | \( 1 - 9.06iT - 169T^{2} \) |
| 17 | \( 1 + 19.7iT - 289T^{2} \) |
| 19 | \( 1 + 25.6iT - 361T^{2} \) |
| 23 | \( 1 - 40.9T + 529T^{2} \) |
| 29 | \( 1 + 22.3T + 841T^{2} \) |
| 31 | \( 1 + 15.9iT - 961T^{2} \) |
| 37 | \( 1 - 44.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 27.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 30.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 65.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 32.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 83.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 72.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 24.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 67.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 30.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 72.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 113. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 103. iT - 9.40e3T^{2} \) |
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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
−9.429956440870972025694510502214, −8.690245627765270189020538597294, −7.41649696825420462424965577080, −7.26674045481896407924661419059, −6.12194757985224661008024349722, −4.99917831476326297433720522441, −4.57027474752118875725555223480, −3.08226150408659900399231204808, −2.15932524337821178428731154786, −0.839397594871898588950500652327,
1.44268912050043574419023529371, 2.74455512146378998610996976239, 3.75762564765636815948322851194, 4.70815191228296500601037581974, 5.47620797040374223816009402427, 6.13909078614159481104648589954, 7.53120206074740766007547616175, 8.116908657961414824165228306641, 9.025850526101602720864638637758, 10.09798126270711117014643230001
