| L(s) = 1 | + (2.67 + 1.36i)3-s + (−3.07 + 5.32i)5-s + (0.511 − 0.295i)7-s + (5.27 + 7.29i)9-s + (−15.1 + 8.72i)11-s + (0.892 − 1.54i)13-s + (−15.4 + 10.0i)15-s − 16.9·17-s − 19.5i·19-s + (1.76 − 0.0911i)21-s + (−6.86 − 3.96i)23-s + (−6.39 − 11.0i)25-s + (4.15 + 26.6i)27-s + (−3.17 − 5.49i)29-s + (27.6 + 15.9i)31-s + ⋯ |
| L(s) = 1 | + (0.890 + 0.454i)3-s + (−0.614 + 1.06i)5-s + (0.0730 − 0.0421i)7-s + (0.586 + 0.810i)9-s + (−1.37 + 0.793i)11-s + (0.0686 − 0.118i)13-s + (−1.03 + 0.668i)15-s − 0.995·17-s − 1.02i·19-s + (0.0842 − 0.00433i)21-s + (−0.298 − 0.172i)23-s + (−0.255 − 0.443i)25-s + (0.153 + 0.988i)27-s + (−0.109 − 0.189i)29-s + (0.892 + 0.515i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.204345127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204345127\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.67 - 1.36i)T \) |
| good | 5 | \( 1 + (3.07 - 5.32i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-0.511 + 0.295i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (15.1 - 8.72i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-0.892 + 1.54i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 16.9T + 289T^{2} \) |
| 19 | \( 1 + 19.5iT - 361T^{2} \) |
| 23 | \( 1 + (6.86 + 3.96i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (3.17 + 5.49i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-27.6 - 15.9i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 58.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (2.66 - 4.62i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-33.9 + 19.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (9.64 - 5.56i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 35.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-20.8 - 12.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-37.9 - 65.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-31.8 - 18.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 87.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 60.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-32.1 + 18.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-66.0 + 38.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 27.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-13.0 - 22.6i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.62971282402842984913374880735, −10.27877106267196460388773303489, −9.135896668497601044774108242067, −8.214191130933229861326396570077, −7.41590550792382591163160948636, −6.76662499751166314741204784153, −5.10441594241300334893814888198, −4.21977742176855298591647211049, −3.02149285955013851294779150962, −2.30544698229947166280731553519,
0.38298187752456921436374068636, 1.91401793398640514668214837655, 3.24058985616766902466638839599, 4.30695094287414122512978188179, 5.37897843461875027744338270188, 6.60442364279360946485474021020, 7.88497900389370048586180812503, 8.214884174892591497138903457121, 8.944062025723192425302093919515, 9.973057835627369603441219663767
