| L(s) = 1 | + (0.0460 − 0.643i)2-s + (−3.14 + 0.683i)3-s + (1.56 + 0.225i)4-s + (1.50 − 1.65i)5-s + (0.295 + 2.05i)6-s + (1.90 − 1.03i)7-s + (0.491 − 2.26i)8-s + (6.66 − 3.04i)9-s + (−0.996 − 1.04i)10-s + (−2.06 − 1.78i)11-s + (−5.07 + 0.363i)12-s + (−1.09 + 2.00i)13-s + (−0.581 − 1.27i)14-s + (−3.58 + 6.22i)15-s + (1.60 + 0.471i)16-s + (3.64 + 2.72i)17-s + ⋯ |
| L(s) = 1 | + (0.0325 − 0.455i)2-s + (−1.81 + 0.394i)3-s + (0.783 + 0.112i)4-s + (0.672 − 0.740i)5-s + (0.120 + 0.838i)6-s + (0.719 − 0.393i)7-s + (0.173 − 0.799i)8-s + (2.22 − 1.01i)9-s + (−0.315 − 0.330i)10-s + (−0.621 − 0.538i)11-s + (−1.46 + 0.104i)12-s + (−0.303 + 0.554i)13-s + (−0.155 − 0.340i)14-s + (−0.926 + 1.60i)15-s + (0.401 + 0.117i)16-s + (0.883 + 0.661i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.797926 - 0.353029i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.797926 - 0.353029i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.50 + 1.65i)T \) |
| 23 | \( 1 + (3.28 + 3.49i)T \) |
| good | 2 | \( 1 + (-0.0460 + 0.643i)T + (-1.97 - 0.284i)T^{2} \) |
| 3 | \( 1 + (3.14 - 0.683i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-1.90 + 1.03i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (2.06 + 1.78i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.09 - 2.00i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-3.64 - 2.72i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.508 - 3.53i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (4.80 - 0.690i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (2.09 - 1.34i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (0.578 + 0.215i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.47 - 3.23i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.96 - 9.03i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-6.88 - 6.88i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.664 + 1.21i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (2.15 + 7.33i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.03 - 4.72i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.96 - 0.140i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-3.58 - 4.13i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (4.79 + 6.40i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-2.46 + 0.722i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.44 - 6.56i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (9.99 + 6.42i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (0.818 + 2.19i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−12.83354487471338010363057878519, −12.26664208837352615674725467996, −11.28221701824081804096176599846, −10.56527499671135606847330853088, −9.794837680771039796602313822125, −7.81561861427196662773778181155, −6.31918629397398597035004176606, −5.50745466669352088734002077246, −4.24229846402223266381122085339, −1.44213433644037167183555519370,
2.07376888410859162511549980845, 5.33371289773756996334936930854, 5.62584517519004817867724309759, 6.99333642474382396802747863503, 7.56871096590202330275996244493, 9.989248799632879336074700258399, 10.79533453918957880410470634456, 11.55666824337247681390193944039, 12.36355640126116519095641338128, 13.66262985803831682140250963840
