| L(s) = 1 | + (1.97 − 0.260i)2-s + (0.814 + 1.65i)3-s + (1.91 − 0.513i)4-s + (−0.751 + 0.659i)5-s + (2.04 + 3.05i)6-s + (−1.29 + 2.30i)7-s + (−0.0277 + 0.0114i)8-s + (−0.238 + 0.311i)9-s + (−1.31 + 1.50i)10-s + (1.59 − 0.104i)11-s + (2.41 + 2.74i)12-s + (3.18 + 3.18i)13-s + (−1.96 + 4.90i)14-s + (−1.70 − 0.704i)15-s + (−3.48 + 2.01i)16-s + (1.29 − 3.91i)17-s + ⋯ |
| L(s) = 1 | + (1.39 − 0.184i)2-s + (0.470 + 0.953i)3-s + (0.958 − 0.256i)4-s + (−0.336 + 0.294i)5-s + (0.833 + 1.24i)6-s + (−0.489 + 0.871i)7-s + (−0.00980 + 0.00406i)8-s + (−0.0795 + 0.103i)9-s + (−0.416 + 0.474i)10-s + (0.480 − 0.0315i)11-s + (0.695 + 0.793i)12-s + (0.884 + 0.884i)13-s + (−0.524 + 1.31i)14-s + (−0.439 − 0.181i)15-s + (−0.872 + 0.503i)16-s + (0.313 − 0.949i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.59209 + 1.80321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.59209 + 1.80321i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.751 - 0.659i)T \) |
| 7 | \( 1 + (1.29 - 2.30i)T \) |
| 17 | \( 1 + (-1.29 + 3.91i)T \) |
| good | 2 | \( 1 + (-1.97 + 0.260i)T + (1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (-0.814 - 1.65i)T + (-1.82 + 2.38i)T^{2} \) |
| 11 | \( 1 + (-1.59 + 0.104i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-3.18 - 3.18i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.0172 - 0.130i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (4.16 + 2.05i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-1.03 - 5.22i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-8.71 + 4.29i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.638 + 9.73i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (1.37 - 6.90i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (4.39 + 10.6i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-2.15 + 8.04i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.302 - 0.394i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (9.30 + 1.22i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-2.05 - 6.05i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (7.93 + 4.57i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.7 - 8.52i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-11.7 - 4.00i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-2.00 + 4.06i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-0.863 + 2.08i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (8.51 + 2.28i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (3.05 - 0.607i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.10096317943857049309255374125, −9.978651638664843124540921276273, −9.133177493262242877125279798033, −8.509448912404749985425385535493, −6.83594013320268665316371696915, −6.11703853866776126305613477457, −5.02114090573983439111039996409, −4.05262146387470933002878164156, −3.47010079166198430164251698094, −2.44823912424744076368867589467,
1.23613743724225662121909144538, 2.98706205250816561737140842603, 3.83097184050631651478528475495, 4.74881911119048822924124974713, 6.17667577596910255391502513923, 6.55558288635542015422288448706, 7.79635222663855598917266559883, 8.281279211541000332893380375786, 9.703779249537320222325274842561, 10.69084878199161437086538152894
