L(s) = 1 | + (0.707 + 0.707i)2-s + (0.575 − 0.575i)3-s + 1.00i·4-s + (−0.185 − 2.22i)5-s + 0.814·6-s + (2.09 − 2.09i)7-s + (−0.707 + 0.707i)8-s + 2.33i·9-s + (1.44 − 1.70i)10-s + 1.39i·11-s + (0.575 + 0.575i)12-s + (4.24 − 4.24i)13-s + 2.96·14-s + (−1.38 − 1.17i)15-s − 1.00·16-s + (−4.38 + 4.38i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.332 − 0.332i)3-s + 0.500i·4-s + (−0.0831 − 0.996i)5-s + 0.332·6-s + (0.792 − 0.792i)7-s + (−0.250 + 0.250i)8-s + 0.779i·9-s + (0.456 − 0.539i)10-s + 0.422i·11-s + (0.166 + 0.166i)12-s + (1.17 − 1.17i)13-s + 0.792·14-s + (−0.358 − 0.303i)15-s − 0.250·16-s + (−1.06 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78843 - 0.00854678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78843 - 0.00854678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.185 + 2.22i)T \) |
| 23 | \( 1 + (-0.664 - 4.74i)T \) |
good | 3 | \( 1 + (-0.575 + 0.575i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.09 + 2.09i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.39iT - 11T^{2} \) |
| 13 | \( 1 + (-4.24 + 4.24i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.38 - 4.38i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 29 | \( 1 + 3.87iT - 29T^{2} \) |
| 31 | \( 1 + 5.74T + 31T^{2} \) |
| 37 | \( 1 + (2.27 - 2.27i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.49T + 41T^{2} \) |
| 43 | \( 1 + (2.85 + 2.85i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.26 + 1.26i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.13 - 4.13i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.66iT - 59T^{2} \) |
| 61 | \( 1 + 13.1iT - 61T^{2} \) |
| 67 | \( 1 + (11.1 - 11.1i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 + (-10.7 + 10.7i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 + (-11.6 - 11.6i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.31T + 89T^{2} \) |
| 97 | \( 1 + (-7.84 + 7.84i)T - 97iT^{2} \) |
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
−12.62746786959135580503668397328, −11.26435552072301645629332382485, −10.48189253204556179855115207865, −8.801345022133054394412383706577, −8.126506102310104290892621361512, −7.41954392408700519536750137265, −5.91194337050684054918545524439, −4.82645011404514831854181383839, −3.84951529013848281097956935373, −1.71284667755214513593035269803,
2.18026424700843169557694441266, 3.44074537551477681456667542450, 4.54828264481667800641502026230, 6.06638786962809521131402961296, 6.90031748097711714709150163755, 8.663142776225284780425198678980, 9.176848498722289623250048577860, 10.61186192375345110846307808967, 11.32381214397072705318629282090, 11.90889539715936109521829860200