| L(s) = 1 | + (0.467 + 1.33i)2-s + (3.31 − 0.373i)3-s + (−1.56 + 1.24i)4-s + (4.08 − 8.47i)5-s + (2.04 + 4.24i)6-s + (−7.80 + 9.79i)7-s + (−2.39 − 1.50i)8-s + (2.05 − 0.468i)9-s + (13.2 + 1.48i)10-s + (−7.87 + 4.94i)11-s + (−4.71 + 4.71i)12-s + (2.62 + 0.599i)13-s + (−16.7 − 5.84i)14-s + (10.3 − 29.5i)15-s + (0.890 − 3.89i)16-s + (−0.855 − 0.855i)17-s + ⋯ |
| L(s) = 1 | + (0.233 + 0.667i)2-s + (1.10 − 0.124i)3-s + (−0.390 + 0.311i)4-s + (0.816 − 1.69i)5-s + (0.340 + 0.707i)6-s + (−1.11 + 1.39i)7-s + (−0.299 − 0.188i)8-s + (0.228 − 0.0521i)9-s + (1.32 + 0.148i)10-s + (−0.715 + 0.449i)11-s + (−0.392 + 0.392i)12-s + (0.201 + 0.0460i)13-s + (−1.19 − 0.417i)14-s + (0.690 − 1.97i)15-s + (0.0556 − 0.243i)16-s + (−0.0503 − 0.0503i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.60551 + 0.385021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60551 + 0.385021i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.467 - 1.33i)T \) |
| 29 | \( 1 + (-28.9 - 1.11i)T \) |
| good | 3 | \( 1 + (-3.31 + 0.373i)T + (8.77 - 2.00i)T^{2} \) |
| 5 | \( 1 + (-4.08 + 8.47i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 + (7.80 - 9.79i)T + (-10.9 - 47.7i)T^{2} \) |
| 11 | \( 1 + (7.87 - 4.94i)T + (52.4 - 109. i)T^{2} \) |
| 13 | \( 1 + (-2.62 - 0.599i)T + (152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (0.855 + 0.855i)T + 289iT^{2} \) |
| 19 | \( 1 + (-0.541 + 4.80i)T + (-351. - 80.3i)T^{2} \) |
| 23 | \( 1 + (-21.0 + 10.1i)T + (329. - 413. i)T^{2} \) |
| 31 | \( 1 + (2.43 + 6.96i)T + (-751. + 599. i)T^{2} \) |
| 37 | \( 1 + (-34.8 - 21.9i)T + (593. + 1.23e3i)T^{2} \) |
| 41 | \( 1 + (-0.911 + 0.911i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (-15.2 - 5.34i)T + (1.44e3 + 1.15e3i)T^{2} \) |
| 47 | \( 1 + (-13.3 - 21.2i)T + (-958. + 1.99e3i)T^{2} \) |
| 53 | \( 1 + (22.3 + 10.7i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 75.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + (55.5 - 6.25i)T + (3.62e3 - 828. i)T^{2} \) |
| 67 | \( 1 + (-51.1 + 11.6i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-20.4 - 4.66i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (5.44 - 15.5i)T + (-4.16e3 - 3.32e3i)T^{2} \) |
| 79 | \( 1 + (-37.6 + 59.8i)T + (-2.70e3 - 5.62e3i)T^{2} \) |
| 83 | \( 1 + (-48.3 - 60.6i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (-39.8 - 113. i)T + (-6.19e3 + 4.93e3i)T^{2} \) |
| 97 | \( 1 + (112. + 12.6i)T + (9.17e3 + 2.09e3i)T^{2} \) |
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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
−15.15314052980646761339923557921, −13.73757259808870188143998105572, −12.96770389285973914054409279882, −12.37270753894453778059796730018, −9.566524048974889199861298089409, −9.014323814847494960235876653154, −8.161739094071448842829022116683, −6.12660349077337335253917079663, −4.96222296922315911988280456126, −2.63472616685861677206263874634,
2.78137771188549494864992651926, 3.53420537264270169010778982473, 6.23483297031683942617738326200, 7.51533464748563468071108162486, 9.392353214570650286537025052127, 10.29805527407063427469174956562, 10.96010229485260266726792303146, 13.13977463424996809977630629845, 13.77857358288355435871704870834, 14.37880695670605432395918043754
