| L(s) = 1 | + (−0.347 − 1.96i)2-s + (−3.39 + 2.85i)3-s + (−3.75 + 1.36i)4-s + (−20.2 − 7.35i)5-s + (6.79 + 5.70i)6-s + (2.98 − 5.17i)7-s + (4 + 6.92i)8-s + (−1.26 + 7.19i)9-s + (−7.46 + 42.3i)10-s + (−13.0 − 22.5i)11-s + (8.87 − 15.3i)12-s + (14.6 + 12.3i)13-s + (−11.2 − 4.08i)14-s + (89.6 − 32.6i)15-s + (12.2 − 10.2i)16-s + (−6.72 − 38.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.654 + 0.549i)3-s + (−0.469 + 0.171i)4-s + (−1.80 − 0.657i)5-s + (0.462 + 0.388i)6-s + (0.161 − 0.279i)7-s + (0.176 + 0.306i)8-s + (−0.0469 + 0.266i)9-s + (−0.236 + 1.33i)10-s + (−0.357 − 0.618i)11-s + (0.213 − 0.369i)12-s + (0.313 + 0.262i)13-s + (−0.214 − 0.0780i)14-s + (1.54 − 0.561i)15-s + (0.191 − 0.160i)16-s + (−0.0959 − 0.544i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0155557 + 0.151987i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0155557 + 0.151987i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 + 1.96i)T \) |
| 19 | \( 1 + (24.7 + 79.0i)T \) |
| good | 3 | \( 1 + (3.39 - 2.85i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (20.2 + 7.35i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-2.98 + 5.17i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (13.0 + 22.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-14.6 - 12.3i)T + (381. + 2.16e3i)T^{2} \) |
| 17 | \( 1 + (6.72 + 38.1i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 23 | \( 1 + (174. - 63.6i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (17.6 - 100. i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (19.6 - 33.9i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-269. + 225. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-130. - 47.5i)T + (6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-49.1 + 278. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + (200. - 73.0i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (90.6 + 514. i)T + (-1.92e5 + 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-6.86 + 2.49i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-133. + 755. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (61.3 + 22.3i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + (719. - 603. i)T + (6.75e4 - 3.83e5i)T^{2} \) |
| 79 | \( 1 + (-320. + 268. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-94.7 + 164. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-304. - 255. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-18.8 - 106. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−15.73957415297467884066600901680, −13.81890544768169657079167234539, −12.33357413336461970221316362219, −11.39798275770990449144606633664, −10.71439125202591375611475344599, −8.841579787063588309550801271846, −7.66521568328970731217594330650, −5.01636241489927416261819341144, −3.84227121714096546829117485068, −0.14511110962621855211362447819,
4.02614759346624285681550504345, 6.11956848942464035214048820937, 7.39137399885387814934146026565, 8.318171674416600245048526293544, 10.49982336067883044804667801747, 11.82451416770356270994167974007, 12.55541998411578965306278524267, 14.53331975278621971243737615545, 15.35767554275643540346245048855, 16.24380315384971790599858786621
