| L(s) = 1 | + (−1.40 − 0.110i)2-s + (1.13 − 3.11i)3-s + (1.97 + 0.311i)4-s + (2.00 + 0.995i)5-s + (−1.93 + 4.26i)6-s + (0.450 − 0.779i)7-s + (−2.75 − 0.657i)8-s + (−6.09 − 5.11i)9-s + (−2.71 − 1.62i)10-s + (3.06 − 1.76i)11-s + (3.20 − 5.79i)12-s + (−1.01 + 0.370i)13-s + (−0.720 + 1.04i)14-s + (5.36 − 5.10i)15-s + (3.80 + 1.23i)16-s + (1.43 + 1.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 − 0.0780i)2-s + (0.653 − 1.79i)3-s + (0.987 + 0.155i)4-s + (0.895 + 0.445i)5-s + (−0.791 + 1.73i)6-s + (0.170 − 0.294i)7-s + (−0.972 − 0.232i)8-s + (−2.03 − 1.70i)9-s + (−0.857 − 0.513i)10-s + (0.922 − 0.532i)11-s + (0.925 − 1.67i)12-s + (−0.282 + 0.102i)13-s + (−0.192 + 0.280i)14-s + (1.38 − 1.31i)15-s + (0.951 + 0.307i)16-s + (0.347 + 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.733607 - 1.01771i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.733607 - 1.01771i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.110i)T \) |
| 5 | \( 1 + (-2.00 - 0.995i)T \) |
| 19 | \( 1 + (1.70 - 4.01i)T \) |
| good | 3 | \( 1 + (-1.13 + 3.11i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-0.450 + 0.779i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.06 + 1.76i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.01 - 0.370i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.43 - 1.70i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.21 + 6.91i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.03 + 3.61i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.929 - 1.61i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.38T + 37T^{2} \) |
| 41 | \( 1 + (2.47 - 6.78i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.0887 - 0.503i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.0815 - 0.0684i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.189 - 1.07i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (3.26 - 2.74i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.0866 + 0.491i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.183 + 0.219i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.58 + 14.6i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (3.01 - 8.28i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-15.6 - 5.67i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.81 - 4.87i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.62 + 9.95i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (10.6 - 8.90i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.05390326394032425114492162115, −10.04758477188063965721238481291, −8.937294366674604885703275962371, −8.332646373722448202926187190484, −7.38135900493106909835840311724, −6.49383718261716827563417487828, −6.08665016432188560161834203116, −3.25882737529989674581781881220, −2.14676326266713261845937200440, −1.17915010564109517304703464564,
2.07207330003830446111993966678, 3.35292211968035839389661694351, 4.84285016507159164223214984058, 5.63988143939171570069852491753, 7.12186628781780390987795413022, 8.553727614409577110925186394952, 9.054241594876983704378078278899, 9.659379702601496302036284946090, 10.28982891233318760316406717780, 11.20519418388558025616655538213
