L(s) = 1 | + (1.10 − 1.33i)3-s + (0.266 − 0.462i)5-s + (2.54 − 0.715i)7-s + (−0.565 − 2.94i)9-s + (−3.39 + 1.96i)11-s + (−0.116 − 0.0674i)13-s + (−0.322 − 0.866i)15-s + 4.32·17-s − 2.22i·19-s + (1.85 − 4.19i)21-s + (−1.70 − 0.983i)23-s + (2.35 + 4.08i)25-s + (−4.55 − 2.49i)27-s + (−5.16 + 2.98i)29-s + (−0.800 − 0.462i)31-s + ⋯ |
L(s) = 1 | + (0.637 − 0.770i)3-s + (0.119 − 0.206i)5-s + (0.962 − 0.270i)7-s + (−0.188 − 0.982i)9-s + (−1.02 + 0.591i)11-s + (−0.0324 − 0.0187i)13-s + (−0.0832 − 0.223i)15-s + 1.04·17-s − 0.511i·19-s + (0.404 − 0.914i)21-s + (−0.355 − 0.205i)23-s + (0.471 + 0.816i)25-s + (−0.877 − 0.480i)27-s + (−0.959 + 0.553i)29-s + (−0.143 − 0.0829i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42257 - 0.716253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42257 - 0.716253i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.10 + 1.33i)T \) |
| 7 | \( 1 + (-2.54 + 0.715i)T \) |
good | 5 | \( 1 + (-0.266 + 0.462i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.39 - 1.96i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.116 + 0.0674i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + 2.22iT - 19T^{2} \) |
| 23 | \( 1 + (1.70 + 0.983i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.16 - 2.98i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.800 + 0.462i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 + (4.59 - 7.95i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.24 - 5.62i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.04 + 5.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 + (1.89 - 3.28i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.35 - 5.39i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.75 - 9.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.22iT - 71T^{2} \) |
| 73 | \( 1 + 0.381iT - 73T^{2} \) |
| 79 | \( 1 + (4.60 + 7.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.28 + 2.21i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 + (-13.6 + 7.89i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.04684166705411811333497121566, −11.03860264091164116229940342951, −9.901665559582197268550732951306, −8.843525779834986090391648642592, −7.78188091838428546558463050183, −7.34458824329298176276614294694, −5.78779608282861589717564680371, −4.59402556642963761441442912456, −2.93295995310648152396624011023, −1.49932994233771599537261324510,
2.25700964571370554301348390799, 3.57381425979581670628485448677, 4.93417646689399348285905001466, 5.80827742551653144000517944814, 7.72435107185328648733226608573, 8.178389113417505178674508563678, 9.324292143331308335186074064259, 10.34423367544022023482606898422, 10.98353631941174752896308674993, 12.07880798573799389176800945532