L(s) = 1 | − 0.368·2-s + 15.5·3-s − 63.8·4-s − 113. i·5-s − 5.74·6-s + 569. i·7-s + 47.1·8-s + 243·9-s + 41.9i·10-s − 814. i·11-s − 995.·12-s + 3.51e3·13-s − 210. i·14-s − 1.77e3i·15-s + 4.06e3·16-s − 7.48e3i·17-s + ⋯ |
L(s) = 1 | − 0.0460·2-s + 0.577·3-s − 0.997·4-s − 0.911i·5-s − 0.0265·6-s + 1.66i·7-s + 0.0920·8-s + 0.333·9-s + 0.0419i·10-s − 0.611i·11-s − 0.576·12-s + 1.59·13-s − 0.0765i·14-s − 0.526i·15-s + 0.993·16-s − 1.52i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.86608 - 0.130762i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86608 - 0.130762i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 15.5T \) |
| 23 | \( 1 + (-1.20e4 - 1.69e3i)T \) |
good | 2 | \( 1 + 0.368T + 64T^{2} \) |
| 5 | \( 1 + 113. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 569. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 814. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.51e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 7.48e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 4.28e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 3.31e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.44e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 7.16e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 7.79e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.13e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 6.28e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.35e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.01e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 3.82e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.29e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 6.45e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + 5.36e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.58e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 1.13e6iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 8.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.25e5iT - 8.32e11T^{2} \) |
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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
−13.40662244386811736842564163938, −12.65821620025433912565860795369, −11.43842285772682047325618599049, −9.509807089755636975561949154071, −8.826152967916648799482275838998, −8.240645063229231477909961993707, −5.88908536842965233169382111671, −4.79166797059639716483985096452, −3.10560147546748576960346051071, −1.03739313377051500381363339996,
1.08776659330188788069541314758, 3.45854959055101569248835641029, 4.35292891145870406726161929510, 6.52571123632975216834463886364, 7.71922113122208140140906365855, 8.880658512597973730503718317109, 10.29865493061060847097298751135, 10.82221458207761763435979736509, 12.94292424947420145556283316456, 13.60188363155945853858605089797