L(s) = 1 | − 2.80i·2-s − i·3-s − 5.85·4-s + 1.80i·5-s − 2.80·6-s + 10.7i·8-s − 9-s + 5.06·10-s + (3.19 − 0.885i)11-s + 5.85i·12-s − 3.43·13-s + 1.80·15-s + 18.5·16-s − 3.48·17-s + 2.80i·18-s + 7.05·19-s + ⋯ |
L(s) = 1 | − 1.98i·2-s − 0.577i·3-s − 2.92·4-s + 0.808i·5-s − 1.14·6-s + 3.81i·8-s − 0.333·9-s + 1.60·10-s + (0.963 − 0.266i)11-s + 1.68i·12-s − 0.951·13-s + 0.466·15-s + 4.63·16-s − 0.844·17-s + 0.660i·18-s + 1.61·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.317095924\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.317095924\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.19 + 0.885i)T \) |
good | 2 | \( 1 + 2.80iT - 2T^{2} \) |
| 5 | \( 1 - 1.80iT - 5T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 + 3.48T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 + 1.36iT - 29T^{2} \) |
| 31 | \( 1 - 4.73iT - 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 + 4.67iT - 43T^{2} \) |
| 47 | \( 1 + 7.09iT - 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 - 6.63iT - 59T^{2} \) |
| 61 | \( 1 - 4.76T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 7.27T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 11.0iT - 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 0.211iT - 89T^{2} \) |
| 97 | \( 1 + 8.21iT - 97T^{2} \) |
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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
−9.202876223991886971942055954110, −8.741169586591937556496537304640, −7.56121563984218107717362311439, −6.74280584200347995777744046652, −5.42416596583203506312251414161, −4.62721562609549422706041450817, −3.43103293596294295310822248275, −2.90146934927466089983533234818, −1.90275384860764405916082139043, −0.77956271813035185459339202628,
0.889364934598624012541212920319, 3.36507014822088712847576192206, 4.52765479310268449333619313624, 4.83310591312994075221478285115, 5.63420445945442660831401050044, 6.61228676814646163518121094521, 7.26395009366900481940397933304, 8.068865404148253948609259317343, 8.921464394622004855669762275060, 9.432570859510691590089639859520