| L(s) = 1 | + (−0.0809 + 0.511i)2-s + (−1.50 − 0.865i)3-s + (1.64 + 0.535i)4-s + (2.21 − 0.305i)5-s + (0.564 − 0.697i)6-s + (0.155 + 0.155i)7-s + (−0.877 + 1.72i)8-s + (1.50 + 2.59i)9-s + (−0.0231 + 1.15i)10-s + (−2.70 − 3.71i)11-s + (−2.00 − 2.22i)12-s + (−2.41 + 0.382i)13-s + (−0.0920 + 0.0669i)14-s + (−3.58 − 1.45i)15-s + (1.99 + 1.44i)16-s + (−2.76 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.0572 + 0.361i)2-s + (−0.866 − 0.499i)3-s + (0.823 + 0.267i)4-s + (0.990 − 0.136i)5-s + (0.230 − 0.284i)6-s + (0.0587 + 0.0587i)7-s + (−0.310 + 0.608i)8-s + (0.500 + 0.865i)9-s + (−0.00732 + 0.366i)10-s + (−0.814 − 1.12i)11-s + (−0.579 − 0.643i)12-s + (−0.669 + 0.106i)13-s + (−0.0246 + 0.0178i)14-s + (−0.926 − 0.376i)15-s + (0.498 + 0.361i)16-s + (−0.671 − 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.904684 + 0.0865741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.904684 + 0.0865741i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.50 + 0.865i)T \) |
| 5 | \( 1 + (-2.21 + 0.305i)T \) |
| good | 2 | \( 1 + (0.0809 - 0.511i)T + (-1.90 - 0.618i)T^{2} \) |
| 7 | \( 1 + (-0.155 - 0.155i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.70 + 3.71i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.41 - 0.382i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (2.76 + 1.40i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (5.97 - 1.94i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.65 - 0.578i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (2.16 - 6.65i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.02 + 3.15i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.311 + 1.96i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-2.06 + 2.84i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-1.96 + 1.96i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.93 - 7.73i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.20 + 2.65i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (3.39 + 2.46i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.06 + 2.22i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (6.01 - 11.7i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-0.166 - 0.0541i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.552 - 3.49i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-10.1 - 3.30i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.78 + 13.3i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (10.0 - 7.31i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-15.8 + 8.07i)T + (57.0 - 78.4i)T^{2} \) |
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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
−14.61298944351563869675191327965, −13.28918249012171768166997216328, −12.51582135085798653991936006077, −11.17536036344697451815226025417, −10.50631041804921384198884698364, −8.695515475972369285213203049627, −7.29832793686903573739809133999, −6.22092509079132667299216331203, −5.29346961530751299926932762980, −2.31203698115944110564499578329,
2.29496789510503299400098423817, 4.76320970291982165362985551247, 6.10396343677158397640787567788, 7.11358072916611544936694254719, 9.391460632004067648932684487350, 10.35544293734645089460739857464, 10.89595743441680231798860870780, 12.27074781273122437950914828605, 13.09463293943697753080421452261, 14.96859758690301846867687114466
