| L(s) = 1 | + (0.654 − 0.755i)2-s + (0.0741 − 0.0476i)3-s + (−0.142 − 0.989i)4-s + (1.57 + 3.45i)5-s + (0.0125 − 0.0872i)6-s + (0.959 − 0.281i)7-s + (−0.841 − 0.540i)8-s + (−1.24 + 2.72i)9-s + (3.64 + 1.07i)10-s + (2.68 + 3.10i)11-s + (−0.0577 − 0.0666i)12-s + (−2.13 − 0.627i)13-s + (0.415 − 0.909i)14-s + (0.281 + 0.181i)15-s + (−0.959 + 0.281i)16-s + (0.664 − 4.62i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.0428 − 0.0275i)3-s + (−0.0711 − 0.494i)4-s + (0.705 + 1.54i)5-s + (0.00512 − 0.0356i)6-s + (0.362 − 0.106i)7-s + (−0.297 − 0.191i)8-s + (−0.414 + 0.907i)9-s + (1.15 + 0.338i)10-s + (0.811 + 0.936i)11-s + (−0.0166 − 0.0192i)12-s + (−0.593 − 0.174i)13-s + (0.111 − 0.243i)14-s + (0.0727 + 0.0467i)15-s + (−0.239 + 0.0704i)16-s + (0.161 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85198 + 0.109208i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85198 + 0.109208i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-4.58 - 1.41i)T \) |
| good | 3 | \( 1 + (-0.0741 + 0.0476i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (-1.57 - 3.45i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-2.68 - 3.10i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.13 + 0.627i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.664 + 4.62i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.773 + 5.37i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.831 + 5.78i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (3.89 + 2.50i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.274 - 0.600i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.69 + 3.70i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-3.70 + 2.37i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 9.37T + 47T^{2} \) |
| 53 | \( 1 + (11.1 - 3.26i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (8.09 + 2.37i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (3.99 + 2.56i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.365 + 0.421i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (3.50 - 4.05i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.13 - 14.8i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-9.49 - 2.78i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (4.35 - 9.54i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-3.91 + 2.51i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (5.11 + 11.1i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.31415645751906888518877305525, −10.98837276261126320172932669088, −9.932652595616887203585265976815, −9.237766388275802317683946451875, −7.45173231787719634523080768755, −6.87801008011663218115699196823, −5.58925119622066415937135738812, −4.53964173308844158033739465206, −2.90654171627731191677930709743, −2.15583839775746401128535120499,
1.40176430173687454086429756090, 3.52131828127895242989254600776, 4.70610938323460333059806288802, 5.71941885290688213093205000423, 6.37074062887133102767959403495, 7.973347257941624209040645115045, 8.899988780539263429791847213725, 9.229239458018469349162406105707, 10.73003343740697587285320831451, 12.15128914148226278014000086965
