L(s) = 1 | + (−5.29 + 9.16i)5-s + (−27.6 − 47.8i)11-s − 83.5·13-s + (−47.6 − 82.4i)17-s + (41.7 − 72.3i)19-s + (−82.8 + 143. i)23-s + (6.5 + 11.2i)25-s − 110.·29-s + (−41.7 − 72.3i)31-s + (−39 + 67.5i)37-s + 412.·41-s + 148·43-s + (232. − 403. i)47-s + (55.2 + 95.6i)53-s + 584.·55-s + ⋯ |
L(s) = 1 | + (−0.473 + 0.819i)5-s + (−0.757 − 1.31i)11-s − 1.78·13-s + (−0.679 − 1.17i)17-s + (0.504 − 0.873i)19-s + (−0.751 + 1.30i)23-s + (0.0520 + 0.0900i)25-s − 0.707·29-s + (−0.241 − 0.419i)31-s + (−0.173 + 0.300i)37-s + 1.57·41-s + 0.524·43-s + (0.722 − 1.25i)47-s + (0.143 + 0.247i)53-s + 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8054262694\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8054262694\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (5.29 - 9.16i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (27.6 + 47.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 83.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + (47.6 + 82.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-41.7 + 72.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (82.8 - 143. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (41.7 + 72.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (39 - 67.5i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 412.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 148T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-232. + 403. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-55.2 - 95.6i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (275. + 476. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (292. - 506. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-130 - 225. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 718.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-334. - 578. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (332 - 575. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 126.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (439. - 760. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−9.243661862041058771988824137197, −8.058517899462671046522971003808, −7.36224066141077091447614818862, −6.98405535563085178967647298880, −5.68037567964221273634321099567, −5.11506746090711710133961207723, −3.95690872144552318200101412250, −2.92259685770942361925310018871, −2.41417295463118537934013478204, −0.55234447022579516666689483540,
0.29425590682355195468331790929, 1.82049762643059120880033544930, 2.59354605374196025154096728909, 4.14829648380211480992168510741, 4.57039782580307964604082970512, 5.41277124188582634451190814692, 6.44477390527989296220484637403, 7.63723710341696827935725341011, 7.74158629434364133183568194976, 8.839053071477709983212461491263