L(s) = 1 | + (0.5 + 0.866i)3-s + (1 − 1.73i)5-s + (−0.499 + 0.866i)9-s + 2·13-s + 1.99·15-s + (3 + 5.19i)17-s + (−2 + 3.46i)19-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s − 0.999·27-s + 6·29-s + (−4 − 6.92i)31-s + (5 − 8.66i)37-s + (1 + 1.73i)39-s + 10·41-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.447 − 0.774i)5-s + (−0.166 + 0.288i)9-s + 0.554·13-s + 0.516·15-s + (0.727 + 1.26i)17-s + (−0.458 + 0.794i)19-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s − 0.192·27-s + 1.11·29-s + (−0.718 − 1.24i)31-s + (0.821 − 1.42i)37-s + (0.160 + 0.277i)39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.054709959\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.054709959\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.643756746774128978901082843881, −9.070808938158782211903376349543, −8.285896089084495706181277932046, −7.59552923938567753712198093543, −6.10708164945966447645064238880, −5.71316423023171652144361350911, −4.47489845484273365129854011737, −3.83132370167834449460302588359, −2.48846889888534038688631328482, −1.19693593535093347729467442026,
1.11080791205412676516311553206, 2.58173054889255842608574208439, 3.17489640671218042275347017375, 4.58150991821872425058945611501, 5.66363381011310389300697430991, 6.56770885847005253148212396806, 7.15454320805606574470453744019, 8.021467268343773864920647447474, 8.990701158314237957478319517873, 9.645969282396918212497116194308