L(s) = 1 | + (−2.01 − 1.16i)2-s + (−1.21 + 1.23i)3-s + (1.71 + 2.97i)4-s + (0.5 − 0.866i)5-s + (3.89 − 1.07i)6-s + (1.11 − 2.39i)7-s − 3.33i·8-s + (−0.0404 − 2.99i)9-s + (−2.01 + 1.16i)10-s + (2.42 − 1.39i)11-s + (−5.75 − 1.50i)12-s − 3.20i·13-s + (−5.04 + 3.53i)14-s + (0.459 + 1.66i)15-s + (−0.459 + 0.795i)16-s + (−0.440 − 0.763i)17-s + ⋯ |
L(s) = 1 | + (−1.42 − 0.824i)2-s + (−0.702 + 0.711i)3-s + (0.858 + 1.48i)4-s + (0.223 − 0.387i)5-s + (1.58 − 0.437i)6-s + (0.422 − 0.906i)7-s − 1.18i·8-s + (−0.0134 − 0.999i)9-s + (−0.638 + 0.368i)10-s + (0.729 − 0.421i)11-s + (−1.66 − 0.433i)12-s − 0.888i·13-s + (−1.34 + 0.946i)14-s + (0.118 + 0.431i)15-s + (−0.114 + 0.198i)16-s + (−0.106 − 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.357162 - 0.283329i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.357162 - 0.283329i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.21 - 1.23i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.11 + 2.39i)T \) |
good | 2 | \( 1 + (2.01 + 1.16i)T + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.42 + 1.39i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.20iT - 13T^{2} \) |
| 17 | \( 1 + (0.440 + 0.763i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.90 - 1.09i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.53 - 3.77i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.15iT - 29T^{2} \) |
| 31 | \( 1 + (7.62 - 4.40i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.203 - 0.352i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.55T + 41T^{2} \) |
| 43 | \( 1 + 0.118T + 43T^{2} \) |
| 47 | \( 1 + (1.31 - 2.27i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.46 + 3.73i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.04 - 3.55i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.7 - 6.17i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.802 - 1.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.25iT - 71T^{2} \) |
| 73 | \( 1 + (-0.192 + 0.110i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.56 + 2.71i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 0.666T + 83T^{2} \) |
| 89 | \( 1 + (0.437 - 0.757i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.37iT - 97T^{2} \) |
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
−13.25198332920463506099148546299, −11.87467397974010327926292399056, −11.18171868532410412064445070943, −10.31778020255620935606129835219, −9.510810936704514405740974871645, −8.494650348796009933501665538266, −7.14335698229331352861591282516, −5.32056021840744485747838014687, −3.58139211027403480765315332422, −1.00903400038061717024357276711,
1.76990431563914865496509141169, 5.29668981685163018397146649296, 6.59157549360708681307352304586, 7.16518846903497013300984959131, 8.556819772962629444442371617431, 9.379909957346089332787866079307, 10.75680932029630261730227354968, 11.60334193393532182014587318765, 12.79406504200886615258345763439, 14.36963016660576444298237501847