| L(s) = 1 | + (1.80 + 2.00i)2-s + (−0.00624 − 0.0993i)3-s + (−0.555 + 5.28i)4-s + (0.993 − 2.36i)5-s + (0.188 − 0.192i)6-s + (−3.43 − 0.288i)7-s + (−7.24 + 5.26i)8-s + (2.96 − 0.374i)9-s + (6.54 − 2.27i)10-s + (−0.100 − 0.0669i)11-s + (0.527 + 0.0221i)12-s + (0.776 − 2.08i)13-s + (−5.62 − 7.41i)14-s + (−0.240 − 0.0839i)15-s + (−13.2 − 2.82i)16-s + (−3.54 − 5.09i)17-s + ⋯ |
| L(s) = 1 | + (1.27 + 1.42i)2-s + (−0.00360 − 0.0573i)3-s + (−0.277 + 2.64i)4-s + (0.444 − 1.05i)5-s + (0.0768 − 0.0784i)6-s + (−1.29 − 0.108i)7-s + (−2.56 + 1.86i)8-s + (0.988 − 0.124i)9-s + (2.07 − 0.720i)10-s + (−0.0303 − 0.0201i)11-s + (0.152 + 0.00638i)12-s + (0.215 − 0.579i)13-s + (−1.50 − 1.98i)14-s + (−0.0622 − 0.0216i)15-s + (−3.32 − 0.705i)16-s + (−0.859 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28899 + 1.44649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28899 + 1.44649i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (11.9 - 2.82i)T \) |
| good | 2 | \( 1 + (-1.80 - 2.00i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (0.00624 + 0.0993i)T + (-2.97 + 0.375i)T^{2} \) |
| 5 | \( 1 + (-0.993 + 2.36i)T + (-3.49 - 3.57i)T^{2} \) |
| 7 | \( 1 + (3.43 + 0.288i)T + (6.90 + 1.16i)T^{2} \) |
| 11 | \( 1 + (0.100 + 0.0669i)T + (4.26 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.776 + 2.08i)T + (-9.84 - 8.49i)T^{2} \) |
| 17 | \( 1 + (3.54 + 5.09i)T + (-5.92 + 15.9i)T^{2} \) |
| 19 | \( 1 + (-4.18 - 3.03i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (6.66 - 2.96i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (-6.22 - 2.46i)T + (21.1 + 19.8i)T^{2} \) |
| 31 | \( 1 + (-0.750 + 0.226i)T + (25.8 - 17.1i)T^{2} \) |
| 37 | \( 1 + (8.57 - 1.45i)T + (34.9 - 12.1i)T^{2} \) |
| 41 | \( 1 + (0.970 + 0.249i)T + (35.9 + 19.7i)T^{2} \) |
| 43 | \( 1 + (-4.69 + 0.394i)T + (42.3 - 7.17i)T^{2} \) |
| 47 | \( 1 + (-0.0303 - 0.108i)T + (-40.2 + 24.3i)T^{2} \) |
| 53 | \( 1 + (-0.515 - 0.812i)T + (-22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (0.929 + 2.86i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (13.2 - 6.55i)T + (36.8 - 48.5i)T^{2} \) |
| 67 | \( 1 + (4.68 + 0.591i)T + (64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (-6.12 + 8.80i)T + (-24.7 - 66.5i)T^{2} \) |
| 73 | \( 1 + (-4.06 - 8.64i)T + (-46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (7.41 - 6.96i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (-5.96 + 3.27i)T + (44.4 - 70.0i)T^{2} \) |
| 89 | \( 1 + (-4.27 - 2.58i)T + (41.2 + 78.8i)T^{2} \) |
| 97 | \( 1 + (-8.21 + 10.8i)T + (-26.0 - 93.4i)T^{2} \) |
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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
−13.57362105940550845216156865616, −12.58399163673883315380313248067, −12.13382515130347788049219584378, −9.890877912527026784129766643046, −8.906226391359765599793387253198, −7.61312188080373407328271628487, −6.65640294773564695472272557100, −5.65529256121138305058062278178, −4.62188381096988025396352992914, −3.35868018867570113870798572377,
2.13727940093271088733393537558, 3.34354675273879527037838450300, 4.42914631642491360745216956349, 6.13842981936834564398088373223, 6.69645335168402436613568982003, 9.309969005645989767344241896858, 10.25083310410635165251678101385, 10.61576532870358577407752123037, 11.93558753411308533319132438388, 12.74207635662942008423460386995
