L(s) = 1 | + (−1 − i)2-s + (−1 + i)3-s + 2i·4-s + (−1 − i)5-s + 2·6-s − 2i·7-s + (2 − 2i)8-s + i·9-s + 2i·10-s + (1 + i)11-s + (−2 − 2i)12-s + (−1 + i)13-s + (−2 + 2i)14-s + 2·15-s − 4·16-s − 2·17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.577 + 0.577i)3-s + i·4-s + (−0.447 − 0.447i)5-s + 0.816·6-s − 0.755i·7-s + (0.707 − 0.707i)8-s + 0.333i·9-s + 0.632i·10-s + (0.301 + 0.301i)11-s + (−0.577 − 0.577i)12-s + (−0.277 + 0.277i)13-s + (−0.534 + 0.534i)14-s + 0.516·15-s − 16-s − 0.485·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.359306 - 0.0714704i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.359306 - 0.0714704i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 5 | \( 1 + (1 + i)T + 5iT^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + (-1 - i)T + 11iT^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + (-3 + 3i)T - 19iT^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-3 + 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-5 - 5i)T + 43iT^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 + (3 + 3i)T + 59iT^{2} \) |
| 61 | \( 1 + (9 - 9i)T - 61iT^{2} \) |
| 67 | \( 1 + (5 - 5i)T - 67iT^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (1 - i)T - 83iT^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−19.49325330313906922783187166826, −17.75343423278250220454729782949, −16.77693473848908654550406060602, −15.82802806170473633912930233477, −13.50004648340396989320504396892, −11.87534745135013107676801105826, −10.79350187367916155153671232394, −9.422303413938473441171880689948, −7.55228244623799316615009824381, −4.38815572194696855867145714346,
5.87627034365529106562658369692, 7.29041351208685469584643073213, 9.036487932412074199687555304026, 10.94615612885111038732160876916, 12.33488696765597057065259773085, 14.43163195458328134002338975151, 15.56248309308998716630007204974, 16.92264916008733610121958362645, 18.21324685159535169645627379191, 18.76920296060781340324975746860