| L(s) = 1 | + (0.496 − 0.0875i)2-s + (−3.57 − 4.26i)3-s + (−3.51 + 1.28i)4-s + (−3.10 − 1.13i)5-s + (−2.14 − 1.80i)6-s + (−2.51 + 4.35i)7-s + (−3.38 + 1.95i)8-s + (−3.81 + 21.6i)9-s + (−1.64 − 0.289i)10-s + (−5.32 − 9.23i)11-s + (18.0 + 10.4i)12-s + (2.66 − 3.18i)13-s + (−0.866 + 2.38i)14-s + (6.28 + 17.2i)15-s + (9.96 − 8.36i)16-s + (−0.355 − 2.01i)17-s + ⋯ |
| L(s) = 1 | + (0.248 − 0.0437i)2-s + (−1.19 − 1.42i)3-s + (−0.879 + 0.320i)4-s + (−0.621 − 0.226i)5-s + (−0.358 − 0.300i)6-s + (−0.359 + 0.621i)7-s + (−0.422 + 0.244i)8-s + (−0.423 + 2.40i)9-s + (−0.164 − 0.0289i)10-s + (−0.484 − 0.839i)11-s + (1.50 + 0.868i)12-s + (0.205 − 0.244i)13-s + (−0.0619 + 0.170i)14-s + (0.419 + 1.15i)15-s + (0.623 − 0.522i)16-s + (−0.0208 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0517i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.439056 + 0.0113776i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.439056 + 0.0113776i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-0.496 + 0.0875i)T + (3.75 - 1.36i)T^{2} \) |
| 3 | \( 1 + (3.57 + 4.26i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (3.10 + 1.13i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (2.51 - 4.35i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (5.32 + 9.23i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-2.66 + 3.18i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (0.355 + 2.01i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (9.04 - 3.29i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-26.8 - 4.73i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (26.7 + 15.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 19.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-40.9 - 48.8i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-62.6 - 22.8i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (4.32 - 24.5i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (-21.2 - 58.4i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (15.5 - 2.74i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (63.3 - 23.0i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-56.9 - 10.0i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-38.6 + 106. i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (74.0 - 62.1i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (31.7 + 37.8i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (48.0 - 83.2i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-31.6 + 37.7i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-38.8 + 6.85i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.55234736136328349940180073550, −10.70077862629877001185493021848, −9.182611941218112355467105955128, −8.103567640820995930120893887726, −7.59659091520183750626563801144, −6.09797014772880794142677194289, −5.63812118021252962339983177638, −4.41326657995177550395230818780, −2.76717601725955514030323084229, −0.75809597940170469101640310404,
0.35621367426243153363015020268, 3.64921559944974973052298825227, 4.23302807432686674974021601985, 5.10617843350851804736043410618, 6.03457770183311034986591815785, 7.21518724874243386865741581111, 8.748667818344522875523225752038, 9.731919814794980585664076231352, 10.28431924928786075235542733074, 10.99289143576023849392788384259
