L(s) = 1 | − 8·2-s + 45.1i·3-s − 45.1i·5-s − 361. i·6-s + (133 + 316. i)7-s + 512·8-s − 1.31e3·9-s + 361. i·10-s + 874·11-s + 2.21e3i·13-s + (−1.06e3 − 2.52e3i)14-s + 2.03e3·15-s − 4.09e3·16-s − 5.96e3i·17-s + 1.04e4·18-s + 3.11e3i·19-s + ⋯ |
L(s) = 1 | − 2-s + 1.67i·3-s − 0.361i·5-s − 1.67i·6-s + (0.387 + 0.921i)7-s + 8-s − 1.79·9-s + 0.361i·10-s + 0.656·11-s + 1.00i·13-s + (−0.387 − 0.921i)14-s + 0.604·15-s − 16-s − 1.21i·17-s + 1.79·18-s + 0.454i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.369522 + 0.556333i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.369522 + 0.556333i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-133 - 316. i)T \) |
good | 2 | \( 1 + 8T + 64T^{2} \) |
| 3 | \( 1 - 45.1iT - 729T^{2} \) |
| 5 | \( 1 + 45.1iT - 1.56e4T^{2} \) |
| 11 | \( 1 - 874T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.21e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 5.96e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 3.11e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 4.73e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.11e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.74e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 3.00e3T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.75e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 3.14e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.24e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 7.64e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 1.13e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.75e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.95e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.84e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 6.09e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 5.34e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 7.14e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 6.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.14e5iT - 8.32e11T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.56593118546249998153867639690, −20.33687755471626900603484997699, −18.67158597206423904655628524524, −17.00653500585425297663577113249, −16.03839533185168513375849669959, −14.42419963223029106500639446527, −11.40316797027950834772575422884, −9.625020434465131307411498781647, −8.798957585047171515468592210723, −4.71547017025821628032419045203,
1.07733282911391212732378645122, 7.01191900691460940820953897559, 8.311442177711380676451943091851, 10.75923984241788276839323529391, 12.86581600670512219937921367609, 14.20134740872490013054366128409, 17.19068626466928186347118288612, 17.82701406890360998772407454060, 19.11798010630834252916645370873, 19.97041013568860581398130146807