L(s) = 1 | + (−0.5 − 1.53i)2-s + (−0.809 − 0.587i)3-s + (−0.5 + 0.363i)4-s + (−0.118 + 0.363i)5-s + (−0.5 + 1.53i)6-s + (2.42 − 1.76i)7-s + (−1.80 − 1.31i)8-s + (0.309 + 0.951i)9-s + 0.618·10-s + 0.618·12-s + (−1.92 − 5.93i)13-s + (−3.92 − 2.85i)14-s + (0.309 − 0.224i)15-s + (−1.50 + 4.61i)16-s + (0.190 − 0.587i)17-s + (1.30 − 0.951i)18-s + ⋯ |
L(s) = 1 | + (−0.353 − 1.08i)2-s + (−0.467 − 0.339i)3-s + (−0.250 + 0.181i)4-s + (−0.0527 + 0.162i)5-s + (−0.204 + 0.628i)6-s + (0.917 − 0.666i)7-s + (−0.639 − 0.464i)8-s + (0.103 + 0.317i)9-s + 0.195·10-s + 0.178·12-s + (−0.534 − 1.64i)13-s + (−1.04 − 0.762i)14-s + (0.0797 − 0.0579i)15-s + (−0.375 + 1.15i)16-s + (0.0463 − 0.142i)17-s + (0.308 − 0.224i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0212164 - 0.892257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0212164 - 0.892257i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 1.53i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (0.118 - 0.363i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-2.42 + 1.76i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (1.92 + 5.93i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.190 + 0.587i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.690 + 0.502i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 + (3.61 - 2.62i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.19 + 3.66i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.42 + 2.48i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.80 - 3.49i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.363i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.28 + 7.02i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.30 + 3.13i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.354 - 1.08i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + (-4.5 + 13.8i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1 + 0.726i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.163 + 0.502i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.92 + 12.0i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 + (-4.64 - 14.2i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.94631369292450859132161787336, −10.41416589307503694930774270563, −9.495343138584048892861775613752, −8.110084534853654804389010326276, −7.38260595263824312404188892283, −6.08403851355427649296377772560, −4.94678110410953670760237073959, −3.50395532119027538868892816777, −2.13314915279618859584762450438, −0.71691118750311535038042708788,
2.19377068025384924919431543777, 4.26101368014768790561735425307, 5.25170572820194156141639314317, 6.19671659381611301634911759590, 7.10175201479451955716167013271, 8.157925404452610575249318927603, 8.892041217218605712615860736684, 9.762190789013834820405709356923, 11.13876108681318117091785551703, 11.79551248016442137915330270671