L(s) = 1 | + (1.62 − 0.606i)3-s + (−0.5 + 0.866i)5-s + (2.62 + 4.54i)7-s + (2.26 − 1.96i)9-s + (−1.33 − 2.31i)11-s + (−1.90 + 3.30i)13-s + (−0.285 + 1.70i)15-s + 3.52·17-s − 4.67·19-s + (7.00 + 5.77i)21-s + (2.47 − 4.29i)23-s + (−0.499 − 0.866i)25-s + (2.48 − 4.56i)27-s + (0.928 + 1.60i)29-s + (4.33 − 7.51i)31-s + ⋯ |
L(s) = 1 | + (0.936 − 0.350i)3-s + (−0.223 + 0.387i)5-s + (0.991 + 1.71i)7-s + (0.754 − 0.655i)9-s + (−0.402 − 0.697i)11-s + (−0.529 + 0.916i)13-s + (−0.0738 + 0.441i)15-s + 0.855·17-s − 1.07·19-s + (1.52 + 1.26i)21-s + (0.516 − 0.895i)23-s + (−0.0999 − 0.173i)25-s + (0.477 − 0.878i)27-s + (0.172 + 0.298i)29-s + (0.778 − 1.34i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82945 + 0.338929i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82945 + 0.338929i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.62 + 0.606i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-2.62 - 4.54i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.33 + 2.31i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.90 - 3.30i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + (-2.47 + 4.29i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.928 - 1.60i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.33 + 7.51i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 + (-1.83 + 3.18i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.76 + 3.05i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.63 + 8.02i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 + (2.10 - 3.63i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.98 - 6.89i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.429 + 0.744i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 6.28T + 73T^{2} \) |
| 79 | \( 1 + (2.81 + 4.87i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.94 - 3.37i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 + (-1.91 - 3.32i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.75245175904221536404784856388, −10.60332149199487401420978828927, −9.370515286003464207888909102874, −8.537291283443477094774994930384, −8.065197574087808681063322032758, −6.82864904086196552248286628299, −5.70118648257996675860530200226, −4.43042167919994180223881376668, −2.87188052615094909976938975548, −2.03622774309509672050065634636,
1.44552161298896703227627031468, 3.20806757218459966568254690268, 4.41094697396850515100216317704, 5.00812274795593065185813647777, 7.05861184820806172923690156181, 7.81702681780078742305705611775, 8.299591938021576109344783424651, 9.749820903908420698910347024531, 10.32906063489754579637168900142, 11.11357463299961939158155160950