L(s) = 1 | + (−1.42 + 0.979i)3-s + (2.60 + 0.456i)7-s + (1.08 − 2.79i)9-s + (0.698 − 0.403i)11-s − 3.86·13-s + (−1.83 + 1.05i)17-s + (−2.70 − 1.56i)19-s + (−4.17 + 1.89i)21-s + (−2.43 + 4.21i)23-s + (1.19 + 5.05i)27-s + 6.67i·29-s + (5.65 − 3.26i)31-s + (−0.603 + 1.25i)33-s + (6.75 + 3.89i)37-s + (5.52 − 3.78i)39-s + ⋯ |
L(s) = 1 | + (−0.824 + 0.565i)3-s + (0.985 + 0.172i)7-s + (0.360 − 0.932i)9-s + (0.210 − 0.121i)11-s − 1.07·13-s + (−0.445 + 0.257i)17-s + (−0.620 − 0.358i)19-s + (−0.910 + 0.414i)21-s + (−0.508 + 0.879i)23-s + (0.229 + 0.973i)27-s + 1.23i·29-s + (1.01 − 0.585i)31-s + (−0.104 + 0.219i)33-s + (1.11 + 0.641i)37-s + (0.884 − 0.606i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8236371607\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8236371607\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.42 - 0.979i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.60 - 0.456i)T \) |
good | 11 | \( 1 + (-0.698 + 0.403i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 + (1.83 - 1.05i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.70 + 1.56i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.43 - 4.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.67iT - 29T^{2} \) |
| 31 | \( 1 + (-5.65 + 3.26i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.75 - 3.89i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.44T + 41T^{2} \) |
| 43 | \( 1 + 0.819iT - 43T^{2} \) |
| 47 | \( 1 + (2.40 + 1.38i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.67 - 11.5i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.86 + 4.95i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.79 - 1.03i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.44 - 5.45i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-1.54 - 2.67i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.76 - 11.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.8iT - 83T^{2} \) |
| 89 | \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 1.01T + 97T^{2} \) |
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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
−9.456073239005001882729439013891, −8.709300960208630333610451176486, −7.84206589163819681899460558790, −6.96135791627677676855825123252, −6.16564506320351014189598145726, −5.25532419745707130632595867927, −4.68550176854216661603129493277, −3.91588562607865554617860837286, −2.58023752394931943440450851165, −1.30971223452550341461920123617,
0.33588541287406802294405680969, 1.72699655966733592182135903745, 2.53764814389902836855256690150, 4.31144425195392275163716501651, 4.70440212526910363037263864177, 5.67080894144932312699620637035, 6.50484421887333921042972734153, 7.20805537361697282440417824081, 7.986447583405163299288242002668, 8.554317015742225538875644789339