L(s) = 1 | + 1.21i·2-s + 1.31i·3-s + 0.525·4-s + (−2.21 − 0.311i)5-s − 1.59·6-s − 2.90i·7-s + 3.06i·8-s + 1.28·9-s + (0.377 − 2.68i)10-s + 0.214·11-s + 0.688i·12-s − i·13-s + 3.52·14-s + (0.407 − 2.90i)15-s − 2.67·16-s − 6.42i·17-s + ⋯ |
L(s) = 1 | + 0.858i·2-s + 0.756i·3-s + 0.262·4-s + (−0.990 − 0.139i)5-s − 0.649·6-s − 1.09i·7-s + 1.08i·8-s + 0.426·9-s + (0.119 − 0.850i)10-s + 0.0646·11-s + 0.198i·12-s − 0.277i·13-s + 0.942·14-s + (0.105 − 0.749i)15-s − 0.668·16-s − 1.55i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.682182 + 0.593037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.682182 + 0.593037i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.21 + 0.311i)T \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 - 1.21iT - 2T^{2} \) |
| 3 | \( 1 - 1.31iT - 3T^{2} \) |
| 7 | \( 1 + 2.90iT - 7T^{2} \) |
| 11 | \( 1 - 0.214T + 11T^{2} \) |
| 17 | \( 1 + 6.42iT - 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 - 4.68iT - 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 + 2.28iT - 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 - 6.36iT - 43T^{2} \) |
| 47 | \( 1 + 1.09iT - 47T^{2} \) |
| 53 | \( 1 + 6.23iT - 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 + 0.280T + 61T^{2} \) |
| 67 | \( 1 - 7.76iT - 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 + 10.2iT - 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 9.52iT - 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 - 18.0iT - 97T^{2} \) |
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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
−15.32903887335426770783810270537, −14.53285538547678360809346543414, −13.14180825694376987059746699447, −11.53510771773781257337920270730, −10.75183512733246721045744708585, −9.311827547440177808522265361757, −7.67243354238388677741909016302, −7.09411328894992678105953640426, −5.12333742502535471730544271856, −3.78286193813014265008327954860,
2.07460690454259973849915542594, 3.90029575512682996202499098165, 6.27684157815518644741708600953, 7.47957343437731933098144318417, 8.812783821490400981211744299532, 10.47341383694129709071912947934, 11.49078673923172472737254844609, 12.47028727162652389162424390978, 12.84374604992164036909273472563, 14.83732895583348212073964348577