| L(s) = 1 | + (0.284 − 1.97i)2-s + (−1.24 − 2.72i)3-s + (−3.83 − 1.12i)4-s + (−10.3 − 11.9i)5-s + (−5.75 + 1.69i)6-s + (−5.39 − 3.46i)7-s + (−3.32 + 7.27i)8-s + (−5.89 + 6.80i)9-s + (−26.5 + 17.0i)10-s + (8.99 + 62.5i)11-s + (1.70 + 11.8i)12-s + (55.2 − 35.5i)13-s + (−8.39 + 9.69i)14-s + (−19.6 + 43.1i)15-s + (13.4 + 8.65i)16-s + (−42.2 + 12.4i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.925 − 1.06i)5-s + (−0.391 + 0.115i)6-s + (−0.291 − 0.187i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.840 + 0.540i)10-s + (0.246 + 1.71i)11-s + (0.0410 + 0.285i)12-s + (1.17 − 0.757i)13-s + (−0.160 + 0.185i)14-s + (−0.338 + 0.742i)15-s + (0.210 + 0.135i)16-s + (−0.602 + 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.174144 + 0.247730i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174144 + 0.247730i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.284 + 1.97i)T \) |
| 3 | \( 1 + (1.24 + 2.72i)T \) |
| 23 | \( 1 + (57.4 - 94.1i)T \) |
| good | 5 | \( 1 + (10.3 + 11.9i)T + (-17.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (5.39 + 3.46i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (-8.99 - 62.5i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (-55.2 + 35.5i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (42.2 - 12.4i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (138. + 40.7i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (25.2 - 7.42i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-102. + 224. i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (251. - 290. i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (85.0 + 98.1i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (38.6 + 84.7i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 - 73.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (515. + 331. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (579. - 372. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-361. + 792. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (80.2 - 558. i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-11.0 + 76.9i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-177. - 52.1i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-184. + 118. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-420. + 485. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (248. + 544. i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (198. + 228. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−12.14831319712831442492123588749, −11.24071862688912744455404752695, −10.06854073676713775482905911658, −8.780732203165268037573305327588, −7.918735574458280819799786141665, −6.50094101540297049718100282324, −4.84473542067951611694343813364, −3.88843042159589435230178656955, −1.76425831656029922067785214807, −0.15075938663838513915589442579,
3.30938289613374563878424347776, 4.21854078498122744637810512726, 6.12370783588735396381558294943, 6.61837005091034776840108529334, 8.257841394272796571240182353273, 8.924527054154443389672111444014, 10.71525193835102865169973315598, 11.08176047551664297521954532172, 12.33722855223977609112037258584, 13.80959953905633698839475105956
