L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.866 + 1.5i)5-s + (−0.5 − 0.866i)7-s + 0.999·8-s + 1.73·10-s + (−0.633 − 1.09i)11-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 0.464·17-s + 4.19·19-s + (−0.866 − 1.49i)20-s + (−0.633 + 1.09i)22-s + (−2.36 + 4.09i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.670i)5-s + (−0.188 − 0.327i)7-s + 0.353·8-s + 0.547·10-s + (−0.191 − 0.331i)11-s + (0.138 − 0.240i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.112·17-s + 0.962·19-s + (−0.193 − 0.335i)20-s + (−0.135 + 0.234i)22-s + (−0.493 + 0.854i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.095221140\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.095221140\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.866 - 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.633 + 1.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 0.464T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.232 - 0.401i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.09 + 5.36i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + (4.73 - 8.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.19 - 7.26i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.09 - 7.09i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + (1.09 - 1.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.69 - 9.86i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.09 + 5.36i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 1.19T + 73T^{2} \) |
| 79 | \( 1 + (2.09 + 3.63i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.36 + 4.09i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.825247504460471900778417444839, −9.300241729895959542475559294367, −8.022534262110833504969761591381, −7.64874564998644970398694626478, −6.61508000701577046692732295399, −5.61256077246342831968727399320, −4.36232878114793167697336750043, −3.41168419399220023942631758140, −2.65832746132345647171870002493, −1.05039393102845973379399064045,
0.69184711285862576741063190169, 2.28256807635847415092220931502, 3.78681490636834353434249576590, 4.78901895873574569084758507151, 5.53053255646544421256820545134, 6.57149960077741543606634293752, 7.32357755153096907012740773385, 8.329913697507933637689284730709, 8.734506418378984844123238437769, 9.683350038617022044367577744716