| L(s) = 1 | − 4.15i·2-s + 494.·4-s + 625·5-s + (−3.75e3 + 5.12e3i)7-s − 4.18e3i·8-s − 2.59e3i·10-s − 3.49e4i·11-s − 3.89e4i·13-s + (2.13e4 + 1.56e4i)14-s + 2.35e5·16-s + 1.12e5·17-s − 3.35e5i·19-s + 3.09e5·20-s − 1.45e5·22-s + 9.47e5i·23-s + ⋯ |
| L(s) = 1 | − 0.183i·2-s + 0.966·4-s + 0.447·5-s + (−0.590 + 0.807i)7-s − 0.361i·8-s − 0.0821i·10-s − 0.719i·11-s − 0.378i·13-s + (0.148 + 0.108i)14-s + 0.899·16-s + 0.327·17-s − 0.590i·19-s + 0.432·20-s − 0.132·22-s + 0.706i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.793495681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.793495681\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 + (3.75e3 - 5.12e3i)T \) |
| good | 2 | \( 1 + 4.15iT - 512T^{2} \) |
| 11 | \( 1 + 3.49e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 3.89e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 1.12e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.35e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 9.47e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 4.08e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 8.08e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + 1.32e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.93e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.20e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.71e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.62e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 9.15e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.25e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 2.14e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.10e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + 8.91e7iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 2.43e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.86e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.75e8iT - 7.60e17T^{2} \) |
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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
−9.934546942609440041802145673283, −9.127206176951515946973929950227, −8.009137063705110255534962134341, −6.90491214640955027259644782285, −6.02010003847087902338704991391, −5.34004968825980270739752721121, −3.51773679449601659847479075102, −2.75259043174573709576140822658, −1.78010153754980051233715086645, −0.51967392454528699313824876076,
1.04234189538211737082684581446, 2.05243052608031061835785810493, 3.13133941401914263616025632945, 4.32364821940357550038574547059, 5.64881566612348874147382216058, 6.61633197000864840738056862064, 7.19312077844128740071471714492, 8.233670672369317773393814565124, 9.590270057186501246434863367577, 10.28119405834040478700363764385
