L(s) = 1 | + (−0.955 + 0.294i)4-s + (−0.623 − 0.781i)7-s + (−1.55 + 1.24i)13-s + (0.826 − 0.563i)16-s + (−1.35 − 0.781i)19-s + (−0.988 − 0.149i)25-s + (0.826 + 0.563i)28-s + (−0.510 + 0.294i)31-s + (−1.82 − 0.563i)37-s + (0.658 + 0.317i)43-s + (−0.222 + 0.974i)49-s + (1.12 − 1.64i)52-s + (−0.400 + 1.29i)61-s + (−0.623 + 0.781i)64-s + (−0.826 − 1.43i)67-s + ⋯ |
L(s) = 1 | + (−0.955 + 0.294i)4-s + (−0.623 − 0.781i)7-s + (−1.55 + 1.24i)13-s + (0.826 − 0.563i)16-s + (−1.35 − 0.781i)19-s + (−0.988 − 0.149i)25-s + (0.826 + 0.563i)28-s + (−0.510 + 0.294i)31-s + (−1.82 − 0.563i)37-s + (0.658 + 0.317i)43-s + (−0.222 + 0.974i)49-s + (1.12 − 1.64i)52-s + (−0.400 + 1.29i)61-s + (−0.623 + 0.781i)64-s + (−0.826 − 1.43i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03662704457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03662704457\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.623 + 0.781i)T \) |
good | 2 | \( 1 + (0.955 - 0.294i)T^{2} \) |
| 5 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (-0.733 - 0.680i)T^{2} \) |
| 13 | \( 1 + (1.55 - 1.24i)T + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 19 | \( 1 + (1.35 + 0.781i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 29 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (0.510 - 0.294i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.82 + 0.563i)T + (0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.658 - 0.317i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (0.826 - 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (0.400 - 1.29i)T + (-0.826 - 0.563i)T^{2} \) |
| 67 | \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (-0.290 + 1.92i)T + (-0.955 - 0.294i)T^{2} \) |
| 79 | \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 - 1.86iT - T^{2} \) |
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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
−9.285202537459896907585351816670, −8.928516194056979783755897098001, −7.69601541956792575137029961892, −7.12326876490299121492575828079, −6.21903540329620238400613616360, −4.90906333092754311942445387534, −4.34983273584192560438597950498, −3.47366781428030281148884648419, −2.12527774224179111093683895933, −0.02943870749834837530785946662,
2.08501646360362318764886522573, 3.26294381228667612873956131774, 4.29067912492607441260120878451, 5.37336825809673311079390856628, 5.76840182699086055063492173057, 6.93612451540163817440942064275, 8.015574611877370563693035438516, 8.597121852866722960107024061396, 9.550154350493744470019830217791, 10.00967600555923561793443100810