L(s) = 1 | + (−0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s − 0.999·21-s + 0.999·27-s + (0.5 + 0.866i)28-s + (−1 + 1.73i)31-s + (−0.499 − 0.866i)36-s + (0.5 − 0.866i)37-s − 0.999·39-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)19-s − 0.999·21-s + 0.999·27-s + (0.5 + 0.866i)28-s + (−1 + 1.73i)31-s + (−0.499 − 0.866i)36-s + (0.5 − 0.866i)37-s − 0.999·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.463357699\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.463357699\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 - T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.879245694950213162958463198032, −8.575494196479600590920330931554, −7.25574859047292966512931792571, −6.77492386386103877983141237002, −5.77351158407256581649650935286, −5.38993879012436745614302960187, −4.41790769047916694597336419642, −3.45058250626535864084691293576, −2.57300080859773070905560532182, −1.49357519845321597025467258940,
1.02593431197738091272503440851, 1.83730551745539479161619287248, 2.91975535002468095560665916862, 3.88309802312457573517559292817, 5.08548430920412413234928432688, 5.86726439754902342590044872300, 6.41134778477984270153572810692, 7.26123255470689522668419821031, 7.926278523403895833945735734969, 8.079147809171169484941832813308