L(s) = 1 | − 4·2-s + (−11.6 − 10.3i)3-s + 16·4-s − 106. i·5-s + (46.7 + 41.2i)6-s + 71.9·7-s − 64·8-s + (30.3 + 241. i)9-s + 424. i·10-s + 484.·11-s + (−187. − 164. i)12-s + 699. i·13-s − 287.·14-s + (−1.09e3 + 1.24e3i)15-s + 256·16-s − 2.02e3i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.749 − 0.661i)3-s + 0.5·4-s − 1.89i·5-s + (0.530 + 0.467i)6-s + 0.554·7-s − 0.353·8-s + (0.124 + 0.992i)9-s + 1.34i·10-s + 1.20·11-s + (−0.374 − 0.330i)12-s + 1.14i·13-s − 0.392·14-s + (−1.25 + 1.42i)15-s + 0.250·16-s − 1.69i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4956147964\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4956147964\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + (11.6 + 10.3i)T \) |
| 59 | \( 1 + (2.62e4 + 5.05e3i)T \) |
good | 5 | \( 1 + 106. iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 71.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 484.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 699. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.02e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 219.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 259. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.94e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.22e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 6.33e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.69e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 7.33e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.84e4iT - 4.18e8T^{2} \) |
| 61 | \( 1 + 5.27e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.08e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.27e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 1.01e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 8.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.31e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.47e5iT - 8.58e9T^{2} \) |
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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
−9.734539399623068914382900632610, −9.098560671386075974790354766019, −8.254921405382007422075904882639, −7.30604417448970167705372879254, −6.24279906192468441164515951857, −5.10713908483718709321357105196, −4.33557821048500786367600536993, −1.87303910393844855267274037050, −1.18532671006261776944496831942, −0.19403974572259381307774953614,
1.55595926043212942220484364431, 3.13374166928655333449782915326, 4.03854111638217484659920261346, 5.85613034905976950085313337195, 6.38294150214518242658773097158, 7.39799583801156260448858580955, 8.448104411491625789423301383375, 9.822491987861063069606263063693, 10.33638491159057227235232076528, 11.07542260935247976485879850828