L(s) = 1 | + (−1.09 + 1.34i)3-s − 0.239·5-s + (−0.619 − 2.93i)9-s − 5.12·11-s + (−2.44 + 4.23i)13-s + (0.260 − 0.321i)15-s + (1.85 − 3.20i)17-s + (1.83 + 3.16i)19-s + 7.42·23-s − 4.94·25-s + (4.62 + 2.36i)27-s + (−1.73 − 3.00i)29-s + (−0.358 − 0.621i)31-s + (5.59 − 6.89i)33-s + (−2.30 − 3.98i)37-s + ⋯ |
L(s) = 1 | + (−0.629 + 0.776i)3-s − 0.106·5-s + (−0.206 − 0.978i)9-s − 1.54·11-s + (−0.677 + 1.17i)13-s + (0.0673 − 0.0830i)15-s + (0.449 − 0.777i)17-s + (0.419 + 0.727i)19-s + 1.54·23-s − 0.988·25-s + (0.890 + 0.455i)27-s + (−0.321 − 0.557i)29-s + (−0.0644 − 0.111i)31-s + (0.973 − 1.20i)33-s + (−0.378 − 0.655i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6791095931\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6791095931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.09 - 1.34i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.239T + 5T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + (2.44 - 4.23i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.85 + 3.20i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.83 - 3.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.42T + 23T^{2} \) |
| 29 | \( 1 + (1.73 + 3.00i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.358 + 0.621i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.30 + 3.98i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.24 + 10.8i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.16 + 3.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.471 + 0.816i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.78 - 6.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.75 + 4.77i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.330 - 0.571i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + (1.83 - 3.16i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.11 + 5.39i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.85 - 8.40i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.74 + 6.48i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.57 + 14.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.414230367742922149717260532849, −8.508460465517126363986737344751, −7.44223467834495514372472397217, −6.90570425482377561937186463906, −5.54046151016659729786977869664, −5.28679515631676867374576947008, −4.28653190837483950952335147241, −3.34774329532091358999179053052, −2.20070132380799128595824935530, −0.32627477053542778284580821855,
1.01150653871650530986157694161, 2.45060064034879559194722941487, 3.22659559251360490006957659093, 4.98115747867143171383974188038, 5.20576468987365390538091285409, 6.18623268560167116915190948657, 7.14428228431155965676925407541, 7.84110165429707193855912870337, 8.222899400067016341275208014390, 9.526520689155298069082531649395