L(s) = 1 | − 4.04e4i·2-s − 1.28e8i·3-s + 6.95e9·4-s + (3.40e11 − 2.60e10i)5-s − 5.19e12·6-s + 4.50e13i·7-s − 6.28e14i·8-s − 1.09e16·9-s + (−1.05e15 − 1.37e16i)10-s + 7.97e16·11-s − 8.94e17i·12-s − 2.76e18i·13-s + 1.81e18·14-s + (−3.35e18 − 4.37e19i)15-s + 3.43e19·16-s + 5.61e19i·17-s + ⋯ |
L(s) = 1 | − 0.435i·2-s − 1.72i·3-s + 0.809·4-s + (0.997 − 0.0764i)5-s − 0.751·6-s + 0.512i·7-s − 0.789i·8-s − 1.97·9-s + (−0.0333 − 0.434i)10-s + 0.523·11-s − 1.39i·12-s − 1.15i·13-s + 0.223·14-s + (−0.131 − 1.71i)15-s + 0.465·16-s + 0.279i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0764i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(2.727027740\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.727027740\) |
\(L(\frac{35}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-3.40e11 + 2.60e10i)T \) |
good | 2 | \( 1 + 4.04e4iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 1.28e8iT - 5.55e15T^{2} \) |
| 7 | \( 1 - 4.50e13iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 7.97e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.76e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 - 5.61e19iT - 4.02e40T^{2} \) |
| 19 | \( 1 + 1.90e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 4.97e22iT - 8.65e44T^{2} \) |
| 29 | \( 1 - 1.05e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 5.13e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 1.64e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 + 3.83e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 6.26e25iT - 8.02e53T^{2} \) |
| 47 | \( 1 - 5.32e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 - 8.17e27iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 2.46e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 5.26e28T + 8.23e58T^{2} \) |
| 67 | \( 1 - 2.38e30iT - 1.82e60T^{2} \) |
| 71 | \( 1 - 2.98e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 8.89e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 - 2.50e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 2.29e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 + 6.67e31T + 2.13e64T^{2} \) |
| 97 | \( 1 + 6.85e32iT - 3.65e65T^{2} \) |
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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
−14.67665416700703084532413933277, −12.96671471089935560076614503743, −12.32393539022460693613928443233, −10.63060231239519384347198888410, −8.453915973014617028239977377399, −6.75862327556656403299195551437, −5.94463569463661448418083900354, −2.67214706485924035350386726647, −1.90115947432172712045360063780, −0.75660764690268216491095841024,
1.99859078826092672289999521555, 3.74073509570432126438788564758, 5.26441652291340084919914627992, 6.65719925957877521660650061533, 8.993490717815409661240755737254, 10.22543639008205003105370843288, 11.34607918386518065239427090259, 14.07554188043248718570606900623, 15.14678228729638862506743559125, 16.51727736855212842569027789654