| L(s) = 1 | + (−1.29 + 1.63i)2-s + (0.470 + 3.12i)3-s + (−0.522 − 2.28i)4-s + (1.19 + 1.74i)5-s + (−5.70 − 3.29i)6-s + (−0.150 + 0.0867i)7-s + (0.651 + 0.313i)8-s + (−6.66 + 2.05i)9-s + (−4.40 − 0.329i)10-s + (2.27 + 0.518i)11-s + (6.89 − 2.70i)12-s + (−0.0203 − 0.271i)13-s + (0.0538 − 0.357i)14-s + (−4.90 + 4.54i)15-s + (2.87 − 1.38i)16-s + (−3.32 + 2.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.919 + 1.15i)2-s + (0.271 + 1.80i)3-s + (−0.261 − 1.14i)4-s + (0.533 + 0.782i)5-s + (−2.32 − 1.34i)6-s + (−0.0567 + 0.0327i)7-s + (0.230 + 0.110i)8-s + (−2.22 + 0.684i)9-s + (−1.39 − 0.104i)10-s + (0.684 + 0.156i)11-s + (1.99 − 0.781i)12-s + (−0.00564 − 0.0753i)13-s + (0.0144 − 0.0955i)14-s + (−1.26 + 1.17i)15-s + (0.717 − 0.345i)16-s + (−0.807 + 0.590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.578838 - 0.474652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.578838 - 0.474652i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (3.32 - 2.43i)T \) |
| 43 | \( 1 + (0.844 - 6.50i)T \) |
| good | 2 | \( 1 + (1.29 - 1.63i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.470 - 3.12i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-1.19 - 1.74i)T + (-1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (0.150 - 0.0867i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.27 - 0.518i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.0203 + 0.271i)T + (-12.8 + 1.93i)T^{2} \) |
| 19 | \( 1 + (4.33 + 1.33i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (-0.898 + 0.968i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.898 + 5.96i)T + (-27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (5.99 - 2.35i)T + (22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (2.20 + 1.27i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.66 - 4.51i)T + (9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-1.10 - 4.84i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (0.308 - 4.11i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (0.413 - 0.199i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (5.98 + 2.34i)T + (44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-8.20 - 2.52i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (-6.52 - 7.02i)T + (-5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-3.72 + 0.279i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-0.471 + 0.271i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-15.2 + 2.30i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (10.8 - 1.63i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-9.48 - 2.16i)T + (87.3 + 42.0i)T^{2} \) |
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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
−10.67174402457316317162578614967, −9.916995745034857690401053394064, −9.272379892272318300243888788383, −8.744767850331760162106380723744, −7.81749792273629676852053458029, −6.48820302605639910494160339908, −6.08631293725267357416270321640, −4.81017338032579944541708467041, −3.83514447668998486616920785920, −2.59312882590171971710765370882,
0.49441557802677931111950908016, 1.64026567159202628554171503761, 2.20189540922755276160293790796, 3.53082539085598528960661651184, 5.38070024534431349366250020283, 6.43705278571747984274573261005, 7.25323127583548975619480157296, 8.373444539855237544266263200566, 8.886166943921473072174228266518, 9.418362694006183731661325622037
