L(s) = 1 | + (0.557 + 1.63i)3-s + (−2.18 − 0.487i)5-s + (−3.90 − 2.25i)7-s + (−2.37 + 1.82i)9-s + (0.631 − 1.09i)11-s + (−4.31 + 2.48i)13-s + (−0.416 − 3.85i)15-s + 4.59i·17-s − 6.38·19-s + (1.51 − 7.65i)21-s + (5.34 − 3.08i)23-s + (4.52 + 2.12i)25-s + (−4.32 − 2.88i)27-s + (1.95 − 3.38i)29-s + (1.08 + 1.88i)31-s + ⋯ |
L(s) = 1 | + (0.321 + 0.946i)3-s + (−0.975 − 0.218i)5-s + (−1.47 − 0.851i)7-s + (−0.792 + 0.609i)9-s + (0.190 − 0.329i)11-s + (−1.19 + 0.690i)13-s + (−0.107 − 0.994i)15-s + 1.11i·17-s − 1.46·19-s + (0.331 − 1.67i)21-s + (1.11 − 0.642i)23-s + (0.904 + 0.425i)25-s + (−0.832 − 0.554i)27-s + (0.362 − 0.628i)29-s + (0.194 + 0.337i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0201960 - 0.154013i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0201960 - 0.154013i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.557 - 1.63i)T \) |
| 5 | \( 1 + (2.18 + 0.487i)T \) |
good | 7 | \( 1 + (3.90 + 2.25i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.631 + 1.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.31 - 2.48i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.59iT - 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 + (-5.34 + 3.08i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.95 + 3.38i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.08 - 1.88i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.516iT - 37T^{2} \) |
| 41 | \( 1 + (-1.11 - 1.93i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.09 + 2.94i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.95 + 3.43i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6.92iT - 53T^{2} \) |
| 59 | \( 1 + (-3.91 - 6.78i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.62 - 11.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.63 - 1.52i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.06T + 71T^{2} \) |
| 73 | \( 1 + 6.90iT - 73T^{2} \) |
| 79 | \( 1 + (-4.09 + 7.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.71 - 3.29i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 0.969T + 89T^{2} \) |
| 97 | \( 1 + (10.8 + 6.24i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.91877673569822506627561481238, −10.71903676130505875261197417119, −10.22985714147554130872780298283, −9.167185187965778943658890526647, −8.432230149195811655722633764256, −7.21285551529371279223585510828, −6.29056436774148250290684753598, −4.59334622570862838962974627733, −3.97137357170745355061546769006, −2.90842271387462293136179937133,
0.094912842253312430140220370829, 2.58333910628983544338462956238, 3.31782494199050439554113822212, 5.05552225919538483041338811291, 6.48584670557253149998958640896, 7.03743283709576863223372490092, 8.032265539902579066719133674341, 9.035502578374753998914486167998, 9.822388453561050355246542609623, 11.19799653939155989368621113644