L(s) = 1 | + (−1.13 − 0.850i)2-s + (0.554 + 1.92i)4-s − 3.87i·5-s + (1.00 − 2.64i)8-s + (−3.29 + 4.37i)10-s − 3.46·11-s + 0.296·13-s + (−3.38 + 2.12i)16-s + 1.56i·17-s + 7.07i·19-s + (7.43 − 2.14i)20-s + (3.92 + 2.95i)22-s − 5.43·23-s − 9.98·25-s + (−0.335 − 0.252i)26-s + ⋯ |
L(s) = 1 | + (−0.799 − 0.601i)2-s + (0.277 + 0.960i)4-s − 1.73i·5-s + (0.356 − 0.934i)8-s + (−1.04 + 1.38i)10-s − 1.04·11-s + 0.0822·13-s + (−0.846 + 0.532i)16-s + 0.380i·17-s + 1.62i·19-s + (1.66 − 0.479i)20-s + (0.835 + 0.628i)22-s − 1.13·23-s − 1.99·25-s + (−0.0657 − 0.0494i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4237977072\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4237977072\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.13 + 0.850i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.87iT - 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 0.296T + 13T^{2} \) |
| 17 | \( 1 - 1.56iT - 17T^{2} \) |
| 19 | \( 1 - 7.07iT - 19T^{2} \) |
| 23 | \( 1 + 5.43T + 23T^{2} \) |
| 29 | \( 1 - 6.85iT - 29T^{2} \) |
| 31 | \( 1 - 2.81iT - 31T^{2} \) |
| 37 | \( 1 - 2.51T + 37T^{2} \) |
| 41 | \( 1 - 3.55iT - 41T^{2} \) |
| 43 | \( 1 + 0.682iT - 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 - 0.623iT - 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 + 2.66T + 61T^{2} \) |
| 67 | \( 1 + 10.6iT - 67T^{2} \) |
| 71 | \( 1 + 0.539T + 71T^{2} \) |
| 73 | \( 1 + 7.39T + 73T^{2} \) |
| 79 | \( 1 + 6.16iT - 79T^{2} \) |
| 83 | \( 1 + 6.15T + 83T^{2} \) |
| 89 | \( 1 - 11.7iT - 89T^{2} \) |
| 97 | \( 1 - 6.84T + 97T^{2} \) |
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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
−9.390557540768188965787162134234, −8.557183776602862867767414724105, −8.150223016449746945469987660067, −7.50539321259143530308797665618, −6.12683144240802500771891128053, −5.26211709093262512876940564599, −4.34252385322894036436133895351, −3.46057839172259880151791853208, −2.04578449131867744961355689480, −1.18383846034101734490577184769,
0.21789538365148923974296670302, 2.28648554425463789002679714797, 2.78225885476151753081976755680, 4.24492931961065317602556587917, 5.47581094765134920836694882881, 6.19059364206543620524952418538, 6.97923623866298845560284040505, 7.50882903351940369071076697691, 8.195951889917667256183611108828, 9.236390708684957392394212205272